
Class 
Book- 



SMITHSONIAN. DEPOSIT 



A QUANTITATIVE STUDY OF 
RHYTHM 



THE EFFECT OF VARIATIONS IN INTENSITY, 
RATE AND DURATION 



HERBERT W^OODROTN^, A.M. 

I<ectnrer in Barrnard College, Columbia University 



SUBMITTED IN PARTIAL FULFILMENT OF THE RE- 
QUIREMENTS FOR THE DEGREE OF DOCTOR OF 
PHILOSOPHY IN THE FACULTY OF PHIL- 
OSOPHY, COLUMBIA UNIVERSITY 



REPRINTED FROM THE ARCHIVES OF PSTCHOLOGX, 

NO. 14 



NEW YORK 

THE SCIENCE PRESS 

1909 



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A QUANTITATIVE STUDY OF 
RHYTHM 



THE EFFECT OF VARIATIONS IN INTENSITY, 
RATE AND DURATION 



BT 

HERBERT T^OODROA^. A.M. 

Liectnrer in Bajrnard Colleg'e, Columbia University 



SUBMITTED IN PARTIAL FULFILMENT OF THE RE- 
QUIREMENTS FOR THE DEGREE OF DOCTOR OF 
PHILOSOPHY IN THE FACULTY OF PHIL- 
OSOPHY, COLUMBIA UNIVERSITY 



REPRINTED FROM THE ARCHIVES OF PSXCHOLOGX, 

NO. 14 



NEW^ YORK 

THE SCIENCE PRESS 

1909 



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CONTENTS 

I. Historical 5 

II. Apparatus and Procedure 12 

III. Intensity 30 

IV. Rate and Intensity 34 

V. Duration 38 

VI. The Meaning of Rhythmical Grouping 53 

VII. Summary 63 



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CHAPTER I 

HiSTOEICAL 

To produce an impression of rhythm, it is necessary to have a 
series of stimuli. These stimuli may be sounds, as in the ease of 
poetry and music, muscular contractions, as in dancing and beat- 
ing time, or lights and electrical shocks, as in some laboratory 
experiments. The stimuli which give the impression of rhythm, 
whatever their nature, may vary in intensity, in duration, and 
in quality, and may be separated by intervals of varying length. 
A fundamental task of the experimental investigation of rhythm 
is to investigate the part played by each of these factors. Only 
after each of them has been studied separately, may we study the 
effect when two or more of them are simultaneously involved, 
and when more complicated factors are introduced, as in melody 
and harmony. 

The aim of the present study is to examine quantitatively the de- 
pendence of the rhythmical impression on the intensity and dura- 
tion of the stimuli. Such an investigation is evidently along the same 
lines as much of the experimental work of Meumann, Bolton, R. 
McDougall, and others, who have studied the objective conditions 
of rhythm. It is necessary, therefore, to review the work that 
has already been done on the perception of rhythm as influenced 
by variations in the intensity and the duration of the stimuli. 

Meumann^ found that in listening to a series of sounds, some 
of which were louder than others, there was a strong tendency 
towards the formation of rhythmical groups. He studied the effect 
of accented sounds on the intervals preceding and following them. 
The most general conclusion at which he arrived is that the effect 
of the more intense sound may be very different according to its 
position in the rhythmical group. ^ He found that sometimes the 
interval following the accented sound is overestimated and some- 
times underestimated, and also that sometimes the interval pre- 
ceding the accented sound is overestimated and sometimes under- 
estimated, and in the cases in which he used more than one sub- 
ject, he gets quite different results under the same objective con- 
ditions. Meumann states, also, that, with most subjects, the sudden 
introduction of a loud sound into a series of weaker ones causes 
an underestimation of the interval preceding, and an overestima- 
tion of the interval following, the loud sound ;^ but he does not 

^ Philos. Stud., g, 264-306, 1894. 
' Ibid., 9, 303 and 10, 311, 1894. 
^ Ibid., g, 276, 1894. 



6 A QUANTITATIVE STUDY OF RHYTHM 

say how many observers gave this introspection, what was the 
introspection of those who did not give it, how many judgments 
were made by each observer, or how they were instructed. Meu- 
mann made no investigation of the effect of duration in rhythm. 

Bolton^ presented sets of sounds of different intensities and 
durations, which recurred always in the same order, and asked 
the subject to point out where the series was grouped. In this 
way, he sought to determine what was the most natural order in 
which the different intensities and durations occurred in the group. 
These experiments led him to state the following general principle: 
"In a series of auditory impressions, any regularly recurrent im- 
pression which is different from the rest, subordinates the other 
impressions to it, in such a way that they fall together in groups. 
If the recurrent difference is one of intensity, the strongest im- 
pression comes first in the group and the weaker ones after. If 
the recurrent difference is one of duration, the longest impression 
comes last."^ 

Bolton calls attention, further, to the long interval which ap- 
peared between the groups, the intervals being objectively equal. 
The pause seemed to be due to the fact that a long interval gener- 
ally preceded the accented sound. At the same time some sub- 
jects, especially 10 and 15, make a short interval after the strongest 
sound. But in another place, Bolton writes: "The accented long 
sound frequently appeared more prolonged than the unaccented 
of the same length: the accent had the effect both to increase the 
length of sound and of the interval which followed." And con- 
sulting his table of results,^ we find that his subjects often found 
the interval preceding the accented sound longer than the others, 
but more often did not. As regards the effect of duration, most 
of Bolton's subjects remarked upon the long interval or pause 
which seemed to follow the long sound, and for this reason it was 
found difficult to make the close of the group come at any other 
place. 

Ettlinger^ has criticized Bolton for his tendency to generalize 
his results on duration, which, being limited to the single case 
in which one sound is twice the duration of the other, do not permit 
of much generalization. So far as they go, however, his results 
indicate that the effect of increasing the length of any regularly 
recurrent sound is to produce an overestimation of the followdng 

^ Amer. J. of Psychol., 6, 222, 1894. 

^ Ibid., p. 232. 

^ Ibid., p. 228. 

* Ztschr. f. Psychol., 22, 132-133, 1900. 



HISTORICAL 7 

interval, while the effect of increasing the intensity is uncertain. 
To most of Bolton's subjects, the strongest sound seemed longer 
than the rest, and the long sound frequently seemed accented. 

Schumann^ asked his subjects to compare the second of two 
intervals enclosed within a series of three sounds with the first 
interval. He found, in the case of four subjects, that when the 
third sound was louder than the preceding, the second interval 
was underestimated as compared with the first. Three of those 
same subjects were also tested with regard to the effect of a loud 
sound which was unexpectedly introduced in a series of weaker 
sounds. In the case of all three, the interval preceding the louder 
sound was apparently shorter than the other intervals. Two 
subjects, on the other hand, obtained the opposite result in both 
experiments, that is, the interval preceding the accent was over- 
estimated as compared with the other intervals. Schumann ex- 
plains this apparent contradiction on the ground that the two last 
mentioned subjects perceived the sounds rhythmically. He made 
no investigation of the effect of the regular recurrence of a more 
intense sound every second or every third time in a long series, 
nor did he study the effect of variation in the durations of sounds 
on rhythm or on the judgment of intervals. 

McDougalP found that a loud sound introduced into a uniform 
series of six beats causes a considerable underestimation of the 
interval following the loud sound, while it less often and less con- 
siderably lengthens the preceding interval. As regards the over- 
estimation of the interval preceding the accent, of the four tables^ 
of results which are presented to prove this, one^ shows an uDder- 
estimation; another^ shows practically no constant error, but an 
underestimation rather than an overestimation, while a third^ 
does not show that accent has any effect on the interval im- 
mediately preceding, but that a longer interval causes over- 
estimation of the interval preceding that longer interval. As 
regards underestimation of the interval following the ac- 
cented sound, we find one table^ which shows that when the 
interval following was 20 per cent, shorter than the interval pre- 
ceding the accent, and 10 per cent, shorter than the remaining 
intervals, it was judged less than the remaining intervals 26 times, 

^ Ztschr. /. Psychol., i8, 30-36, 1898. 

^ Harvard Psychol. Stud., i; Monog. Sup. Psychol. Rev., 4, 309-412, 1903. 

3 Op. cit.. Tables XXVIII, XXIX, XXXI and XXXII. 

4 Table XXIX. 
s Table XXXI. 

« Table XXXII. 
7 Table XXIX. 



8 A QUANTITATIVE STUDY OF RHYTHM 

but as equal to them 31 times, and greater six times. The table 
shows, therefore, that the interval following the accented sound, 
which was 10 per cent, less than any of the other intervals, was 
judged either equal or greater in 59 per cent, of the cases. This 
is about as strong evidence of overestimation as the other three 
tables give of underestimation. Still more doubt is thrown upon 
McDougall's conclusion in this connection from the fact that, though 
five subjects are said to have participated in the experiments repre- 
sented in these four tables, no separate record is kept of these 
individuals, and even then the total number of judgments on any- 
one set of intervals is very often not over six. These experiments 
were made as already stated with series of six sounds, one of which 
was louder than the others. Other experiments were made by 
McDougall in which series which might be called rhythmical were 
presented to the subject, that is, series in which the accented sound 
recurred regularly every other time or every third time. In these, 
it is quite evident that the interval preceding the accented sound 
is overestimated as compared with the other intervals. McDougall 
moreover determines the magnitude of this relative overestimation 
of the interval preceding accent in rhythmical groups of both two 
and three sounds. To accomplish this, he ascertained at what 
temporal spacing the grouping disappeared, that is, what relative 
length of the intervals before and after the accented sound was 
necessary to produce the impression of temporal uniformity in the 
series. He refers to the point at which temporal uniformity takes 
the place of rhythmical grouping as the indifference point. Con- 
cerning this point, he writes: "At a certain definite stage in the 
process the tendencies toward the two forms of apprehension bal- 
ance each other, so that with the slightest change in direction of 
attention the rhythmical figure inverts and reverts to the original 
form indifferently."^ As regards the effect of duration, McDougall 
made no investigations corresponding to those on intensity. 

Miner^ investigated the effect of intensity and of duration in 
visual rhythms, using lights in place of sounds. His subjects mis- 
took a difference in the duration of the lights for one of intensity. 
The more intense light regularly recurred every second or every 
third time. The intervals between the lights were always equal. 
The subjects were asked to group the lights first, in groups in which 
the brighter came first, and second, in groups in which the brighter 
came second. While doing this, they had, further, to judge which 
interval was the longer, and which the shorter. We have no guaran- 

^ Op. cit., p. 382. 

* Monog. Sup. Psychol. Rev., 5, No. 4, 1903. 



HISTORICAL 9 

tee that the subjects actually held to this forced rhythm while ob- 
serving the relative length of the intervals. In fact, Miner, him- 
self, distinguishes two attitudes on the part of these subjects, 
though all were similarly instructed. These attitudes he calls the 
rhythmical and non-rhythmical. He reports no introspections, 
however, on the part of the subjects, on which to base this division 
of them into rhythmical and non-rhythmical, and it is made 
merely as an explanation of the disagreement between the results 
of different individuals. His results do not admit of much gen- 
eralization. The following statements, however, can be made. 
The interval which was most often judged the longest, no matter 
how the sounds were grouped, was that which separated the groups. 
The interval next most frequently judged the longest was that 
before the brighter light. Like Meumann, Schumann, and Mc- 
Dougall, Miner also investigated the effect of only one intense 
stimulus in a longer series of less intense. He found that twelve 
out of seventeen subjects judged the interval after the bright flash 
to be the longer. 

Miner says that the discrepancy between his own results and 
those of McDougall may be due to the difference in the quality 
of the stimulus or the length of the intervals. But this would not 
explain the discrepancy between Meumann's statements and those 
of McDougall. Of more importance, probably, is the fact that, 
in Miner's experiments, the subjects did not know at what point 
in the series to expect the intense stimulus, whereas, in McDougall's, 
they did. Miner says that the bright flash was brought in at an un- 
expected time. Moreover, the subjects were told to notice any 
difference in the appearance of the interval or the light that followed 
the bright flash, that is, their attention was directed to the interval 
and stimulus following the more intense stimulus. With such 
instructions, we do not know whether the interval following the 
brighter light appeared longer because of the brighter light or be- 
cause the subject's attention had been specially directed to the 
interval and light following the bright flash. McDougall's sub- 
jects, on the other hand, knew when the loudest sound was to ap- 
pear. He writes: "As a single hearing very commonly produced 
but a confused impression due to what was reported as a condi- 
tion of unpreparedness the method adopted was 

to repeat each series before asking for a judgment 

In order to define the direction of attention on the part of the ob- 
:server, it was made known that the factors to be compared were 
the durations of the intervals adjacent to the louder sound in re- 



10 A QUANTITATIVE STUDY OF RHYTHM 

lation to the remaining intervals of the series."^ This difference 
in the instructions given the subjects may account for the differ- 
ence in the results. In other words, the disagreement between 
the results of Miner and McDougall may be interpreted as merel}^ 
a piece of evidence of the importance of the direction of attention 
in the estimation of intervals. The results of both investigations 
agree in that they find an overestimation of the interval preceding 
that stimulus which, there is reason to suppose, is the object of 
the greatest amount of expectant attention, namely, the more 
intense, when the subject knows when to expect it,^ or the one after 
the more intense, when they are not informed when to expect the 
more intense, but have had their attention especially directed, 
as a result of their instructions, to the stimulus following the more 
intense. The apparent discrepancy between the results of Miner 
and McDougall, therefore, goes to confirm Schumann's theory 
concerning the effect of strain of attention in the estimation of 
small intervals. 

Whether Miner's results are to be considered as contradicting 
those of Bolton on the effect of an increase in duration is not very 
clear. Miner found that his subjects mistook an increase in the 
duration of a light for an increase in intensity, and finds that this 
apparent increase in intensity has about the same effect on the 
rhythmical grouping as an actual objective increase in the intensity 
of the stimulus. But as we have already pointed out. Miner's 
experiments do not permit us to generalize concerning this effect. 
And, in fact, we have no way of being sure which of the large num- 
ber of varieties of effect he mentions were due to apparent intensity 
and which to the attempt of the subjects to group the sounds in 
the way that they were instructed to group them. 

Notwithstanding many apparent contradictions among the in- 
vestigations I have reviewed, it is yet possible to indicate the general 
trend of results so far obtained concerning the effect of variations 
in the duration and intensity of certain stimuli in a series. There 

^ Monog. Sup. Psychol. Rev., 4, 362-363, 1903. 

^ G. F.°Arps und O. Klemm, "Der Verlauf der Aufmerksamkeit bei ryth- 
mischen Reizen." Psychol. Stud., 4, 518-528, 1909. These authors, using the 
sensitivity to temporal displacements of each of the members of a dactylic 
group (every 3rd sound of the objective series being accented and the intervals 
all equal) as a measure of the degree of attention bestowed on each of the 
members, conclude that the greatest degree of the attention occurs at the 
accented sound and the least at the second unaccented sound. The greatest 
change in the level of attention occurs therefore during the interval preceding 
the accented sound. Apart from this work we have the generally recognized 
fact that intensity is one of the "objective conditions" of attention. 



HISTORICAL 11 

can be no question but that the effect of a more intense stimulus 
in a series of less intense is different, according as the more intense 
occurs unexpectedly, or at a time when it is expected, as when 
the more intense stimulus regularly recurs. The general con- 
clusion indicated, if I may neglect, for the sake of simplicity, the 
special conditions of the above-mentioned researches, is that a more 
intense stimulus, if unexpected, causes a relative underestima- 
tion of the interval preceding it; if expected (or regularly recurrent), 
a relative underestimation of the interval following it. 

As regards duration, about all that can be said, neglecting special 
conditions again, is that in the case of regularly recurrent differ- 
ences, one investigator has found an overestimation of the interval 
preceding the longer stimulus, another, indications of an overes- 
timation of the interval following the longer stimulus. 



CHAPTER II 

Apparatus and Procedure 

The following experiments were performed in the psychological 
laboratory of the University of Michigan, under the guidance of 
Professor Pillsbury and Dr. Shepard, whom I am glad to thank 
for their advice and aid. The work was done mostly during the 
summer vacations of 1907 and 1908. The subjects used, thirteen 
in all, were, with one exception, advanced students doing original 
work in experimental psychology. I take this occasion to express 
to all those who acted as subjects my appreciation of their patience 
and conscientiousness. 

The first question to be solved, in an investigation of this sort, 
is how to produce a series of sounds in which it is possible to make 
accurately measurable variations in the absolute and relative length 
of the intervals between the sounds, their absolute and relative 
intensity, their absolute and relative duration, and the relative 
proportion of sound and silence. I will describe, first, the dispo- 
sition of apparatus for work in which the effect of intensity was 
studied and then point out the changes that were necessary for 
the study of variations in the duration of the sounds and their rate 
of succession. 

Part of the apparatus used in the production of the series of 
intensively changeable sounds is shown in the cut, opposite page 13, 
which represents the operator's room. In addition, there was 
in the subject's room a telephone, head-rest, and a microscope 
for reading the amplitude of vibration of the telephone plate. This 
reading was not used in the final estimate of the intensity of the 
sound, but merely as a preliminary guide. 

The sound used for producing the rhythm was that made by a 
telephone receiver, through which passed an alternating current 
of 60 alternations per second (a branch from the city lighting cir- 
cuit), which current, in turn, was interrupted 250 times per second 
by a tuning fork. The object in having the current interrupted 
by a tuning fork was to get rid of a click which otherwise occurred 
at the moment of breaking the current. It was found that, if a 
current which produced an approximately pure tone was used, 
there was a very slight click, both at the instant of making the 
circuit and of breaking it, especially, however, at the break. One 
could not be sure in judging by the ear alone that this click was 
not an illusion due to contrast with the preceding or following 
silence; but Professor Pillsbury succeeded in demonstrating its 



APPARATUS AND PROCEDURE 



13 



existence objectively through the aid of a string galvanometer 
set up in circuit with a microphone placed in front of the telephone. 
I tried many methods of eliminating this click, suggested by officers 
of the departments of physics and of electrical engineering, but 
they all failed. Finally, I noticed that in some experiments in 
which I was using an induction coil in order to get a telephone 
sound, there was no click at the break and make. As the ordinary 
inductorium is not absolutely reliable as regards the regularity 




T 

Fig. I 

M = Meumann's machine. 

C1C2 = contacts. 

S1S2 = adjustable shunt resistances. 

RjRj = adjustable resistances. 

WW = wall. 

T = telephone. 

F = 250 fork. 

of vibration and resistance of its interrupter, at the suggestion of 
Dr. Shepard, a tuning fork with a vibration frequency of 250 per 



14 A QUANTITATIVE STUDY OF RHYTHM 

second was substituted. The platinum plate of the fork was pro- 
vided with a specially devised micrometer screw arrangement, 
which made it possible accurately to control the closeness of the 
contact. Needless to say, this contact, as well as all others, were 
kept bright as new throughout the investigation and frequently 
renewed. The sound produced in this manner, according to all 
the subjects who took part in the investigation, was perfectly even 
and uniform, and was absolutely free from a click at either the 
beginning or ending. 

For the purpose of making and interrupting the sounds, a Meu- 
mann's time-sense apparatus' was used, on which were arranged 
ordinarily from 2 to 6 contacts. Each contact, at the time it was 
closed, formed part of a separate circuit. By changing the amount 
of resistance in these separate circuits, the intensity of any sound 
could be varied independently of that of any other. This arrange- 
ment is shown in the diagram on the preceding page. 

Whenever any measurements of intensity were made, all the 
contacts and all circuits were arranged as in actual use. This was 
necessary inasmuch as part of each circuit constituted a shunt for 
the other circuits. 

As a measure of the intensity of the sounds, the distance to which 
they were just audible was taken. It is not claimed that such a 
method of measurement is very exact, but on the whole it was con- 
sidered the most satisfactory. The measurements were all made 
on the same day, Sunday, during the sum.mer vacation, about two 
weeks before the opening of college, in Ann Arbor, in an open space 
on the campus left by the removal of an old building. Only a few 
series of measurements were taken for each sound, as Sunday was 
the only day it was quiet enough to work, and it was found impossible 
in the course of about two months to get more than one good Sun- 
day, that is, one which was free from wind or rain, or the noise 
of birds, crickets, etc. I obtained as many series of measurements 
as was possible in one day, working from early in the morning until 
after dark, with Mr. Dockeray, at the time assistant in psychology 
in the University of Michigan laboratory. Environmental con- 
ditions were constant throughout the day. The method of minimal 
changes was used, and from 2 to 6 series were obtained with each 
intensity. At each step, the sound was presented three times in 
succession, each time for about 1 second. The operator gave 
the subject a signal by waving a handkerchief a short but variable 
time before the first sound, and the subject indicated his judgment 
by raising a handkerchief whenever he thought he heard the sounds 

^ Phil. Stud., 12, 142-152, i8g6 



APPARATUS AND PROCEDURE 15 

and keeping it lowered the rest of the time. The intervals between 
the three successive presentations of the sounds were made quite 
irregular. If the subject signaled at such times as to indicate that 
he heard all three sounds or two out of three, he was considered 
to have judged "sound audible;" if he signaled correctly for only- 
one of the three sounds, his judgment was called "doubtful;" if 
he got none right, he was marked as if he had judged "sound inaudi- 
ble." The method used was therefore really a combination of the 
method of minimal changes and the method of constant stimuli. 
The following measurements were made of the sounds used in 
this investigation, the distances being given in feet: 



Distance audible 


No. of series 


M. v. (absolute 


24 


6 


I .2 


28 


2 


I.O 


30 


2 


0.8 


32 


4 


2.6 


40 


2 


2.0 


70 


3 


1.2 


136 


3 


4-3 


196 


2 


6.0 


300 


4 


24.6 


420 


4 


17-3 


616 


2 


35-0 


800 


4 


56.0 


1 100 


2 


50.0 



In the remainder of this work, whenever the intensity of a sound 
is indicated, what is meant is the distance to which it was audible. 

The rate of rotation of the Meumann's apparatus, and so the 
rate of succession of the sounds, was controlled by a Helmholtz 
motor and two ball-bearing speed-reducers. The rate of rotation 
of the Helmholtz motor was kept as constant as possible through 
the aid of a speed counter and stop watch, and an adjustable re- 
sistance in the circuit passing through the motor. An accurate 
record of the rate of rotation as well as the duration of each sound 
and each interval could be obtained by placing in the same circuit 
with the telephone a time marker writing on a drum alongside 
another time marker of 100 single vibrations per second. It was, 
of course, out of the question to measure individually every one 
of the sounds and intervals used, as the total number was over 
half a million. What was done was to take several drums of records 
of each different rhythm used, both at the beginning and end of 
the hour, and to keep testing the rate of revolution of the time- 
sense apparatus by a stop-watch and speed counter during the 



16 A QUANTITATIVE STUDY OF RHYTHM 

hour. How great a degree of regularity of speed was obtained 
is indicated by the mean variation of the following measurements 
of the times of successive revolutions: 

Duration of i revolution N. M. V. per cent. 

1.50 sees. 50 0.3 

3.00 sees. 56 0.5 

A part of this variation is, of course, due to inaccuracies in esti- 
mating fractions of a vibration of the fork. I will show later (Chap. 
IV) that my results would not be changed even by very consid- 
erable variations in the rate of revolution. Such extremely small 
mean variations in the rate of revolution as those just indicated 
have, therefore, no significance for the results of this investigation, 
and for that reason, in the tables to follow, no mention is made 
of the mean variation in the rate of revolution of the time-sense 
apparatus. Changes in the rate were made by the aid of the speed- 
reducers. A range of rate was obtained from one revolution in 
0.5 seconds to one revolution in 26.0 seconds. 

When variations in the duration of the sounds were desired, 
contacts of different length were used. The accurate measure- 
ment of the duration of the sounds was accomplished in the manner- 
described above. Finally, it was possible, within certain limits, 
to arrange for any desired combination of intensity and duration 
in the sounds composing the rhythmical series. 

Given a series of sounds which may be varied in intensity and 
duration, the question arises, how are we to investigate the effect 
of these variations? The question to be solved is, what values, 
have intensity and duration for the impression of rhythm which 
is obtained in listening to the series of sounds? If the study is 
to take on a quantitative aspect, there must be some measure of 
the magnitude of the rhythmical attribute of the total impression;, 
there must be some index to show us whether more or less rhythm 
is felt, some way of telling which of two rhythms, both of which 
may perhaps be qualitatively alike, in the sense that they are both 
trochaic or both iambic, is the stronger or more emphatic. The 
members of a group may be thought of as being held together more 
or less securely, by stronger or weaker bonds; and what we desire 
is a measure of the force with which these members of a group are 
held together — a measure of what McDougall calls "the rhythmic 
integration of the stimuli." 

A very direct method would be to ask the subject to introspect 
regarding the relative amount of rhythm produced by two different 
series of sounds. This task is similar to that required of subjects 



APPARATUS AND PROCEDURE 17 

in experiments on the so-called intensity of sensation. The intro- 
spection in the case of rhythm, however, is more difficult, and I 
doubt if reliable results can be obtained from an investigation car- 
ried out in this way. For instance, suppose we compare two trochaic 
rhythms, one produced by an alternation in intensity and one by 
an alternation in duration. It may be very apparent that both 
rhythms are what we call trochaic, but the total impression is very 
different in the two cases, and it is very hard to isolate the intensive 
aspect; and consequently it is difficult to say in which case the 
rhythm is the stronger; and it would be still more difficult to say 
which is the stronger if one were trochaic and the other iambic. 
I found that such judgments were not impossible, but I have pre- 
ferred a more indirect and more objective method, one that de- 
mands something far less difficult on the part of the subject. 

The following description of the method used in this research 
may be found rather difficult to follow by those who are unac- 
customed to the terminology of rhythm; but I believe that, if the 
reader will take a pencil and paper and follow out by the aid of 
symbols the procedure below outlined, he will find no ambiguities. 
It should be remembered that in all the rhythms here dealt with 
every second or every third sound is either louder or longer than 
the others. Also that rhythm is characterized by an apprehension 
of the sounds in groups and that when there is no grouping there 
is no rhythm. The method consists in taking as the measure of 
the amount of rhythm the amount by which, measuring from the 
point at which all intervals are equal, the internal intervals, or 
intervals within the group, must be increased or decreased with 
respect to the external intervals, or the intervals between the groups, 
in order to cause a disappearance of the rhythm, that is, a disap- 
pearance of apparent grouping. Roughly speaking, I have used 
as the measure of the amount of rhythm the amount of work that 
had to be done on the intervals to destroy the rhythm. Suppose, 
for instance, that a certain series of sounds in which every other 
one is accented, produces an impression of trochaic rhythm, the 
accented sound seeming to begin the group. Now, such a rhythm 
can be changed to an iambic one by increasing the interval fol- 
lowing the accented sound, with respect to the interval preceding 
the accented sound. I have found no exception to this possibility. 
Moreover, as the process of increasing the interval after the accented 
sound is going on, just before the rhythm becomes iambic, there 
will be a point reached at which the rhythm can hardly be said 
to be more iambic than it is trochaic. This point may be called 
the iambic-trochaic indifference point. It is the point at which 



18 A QUANTITATIVE STUDY OF RHYTHM 

the rhythm is destroyed, or at least reduced to a minimum. If, 
to arrive at this point, it was necessary to produce only a very 
slight increase in the duration of the interval after the louder 
sound with respect to the duration of the interval before it, the 
rhythm may be said to have possessed only a slight degree of tem- 
poral segregration to begin with; and that slight degree of temporal 
segregation to have been in the trochaic direction. This means 
that, to begin with, the temporal grouping of the sounds was such 
that the sounds within the same group were only slightly 
more grouped than two successive sounds belonging to different 
groups. 

Now, if it is desired to determine the effect of intensity or duration 
on temporal segregation, we must be able to separate the influence 
toward temporal segregation exerted by intensity and duration 
from that exerted directly by unequal temporal spacing of the 
sounds. The temporal segregation we wish to measure is, of course, 
the apparent, or subjective, temporal segregation — the tem- 
poral segregation presented by the rhythm consciousness and not 
that presented by the objective series. If the indifference point 
occurs where the intervals are objectively equal, then any differ- 
ences in duration or intensity which may exist in the sound series 
are evidently exerting no influence towards temporal segregation. 
To measure the amount of temporal segregation produced by changes 
in intensity or duration we must determine the amount of change 
in the intervals, from objective equality, necessary to arrive at the 
indifference point, or, what amounts to the same thing, take as 
the measure of the temporal segregation produced by intensity 
or duration the difference at the rhythm indifference point be- 
tween the intervals before and after the accented or the longer sound. 
If, when the intervals are equal and every other sound accented, 
the rhythm is heard as trochaic, but if this trochaism^ can be de- 
stroyed by increasing the interval after the accented sound, then 
it is clear that the accent is exerting some influence toward tem- 
poral segregation in the trochaic direction. Moreover, if in one 
case the interval after the accented sound has to be increased 10 
per cent., starting with intervals equal, and in a second only 1 per 
cent., in order to arrive at the point of indifference, it is clear that 
a greater degree of temporal segregation in the trochaic direction 
exists in the first case than in the second. On the other hand, 
if, with equal intervals, a series of sounds should produce the im- 
pression of iambic rhythm, but this impression is changed to one 

^ This word was suggested to me by Professor Titchener as at least prefer- 
able to "trochaicness." 



APPARATUS AND PROCEDURE 19 

of indifference by increasing by 10 per cent, the interval before 
the accented sound, then this rhythm could be said to present the 
same amount of temporal segregation as the series which, with 
equal intervals, produced the impression of trochaic rhythm, but 
in which an increase of 10 per cent, in the interval after the accented 
sound brought the listener to the indifference point. Both rhythms 
could be said to present the same degree of temporal segregation; 
but one would be in the iambic direction and the other in the trochaic. 
The influence towards temporal segregation exerted by any such 
factor as recurrent differences in accent, pitch, duration, etc., may, 
therefore, be measured by the difference between the external 
and internal intervals of the group at the indifference point. In 
the tables of the following chapters this difference has been re- 
corded in the columns headed A-B, which means the difference 
in duration at the rhythm indifference point between the interval 
after the accented or the longer second (A) and the interval before 
the accented or longer sound (B). When A-B is positive, this 
means that the series is heard as trochaic, when negative, as iambic, 
providing the intervals are objectively equal. Further, the magni- 
tude of A-B, when positive, is a measure of the degree of temporal 
segregation in the trochaic sense; when negative, of the degree 
of temporal segregation in the iambic sense — so far as this segrega- 
tion or grouping is due to other factors than objective difference 
of intervals. 

But is measuring the amount of temporal segregation the equiv- 
alent of measuring the amount of rhythm, the amount of trochaism 
or iambism? I shall show, in Chapter VI, that rhythmical group- 
ing is a temporal grouping. Rhythmical segregation implies (sub- 
jective) temporal segregation. Then "more" or "less" applied 
to rhythm means more or less temporal segregation; and, in meas- 
uring the amount of temporal segregation, we obtain an index 
of the quantity of rhythm. This same conclusion may be reached 
by a different line of reasoning. The judgments, iambic, trochaic 
and doubtful, have a certain range of distribution. If we admit 
that a rhythm which is judged trochaic 95 per cent, of the judg- 
ments is more trochaic than one which is judged trochaic in 60 per 
cent, of the judgments, then we must also admit that our measure 
of temporal segregation serves also as a measure of more or less 
rhythm: because as the degree of temporal segregation in the trochaic 
direction exerted by an alternation of more and less intense sounds 
increases (because of increase in the ratio between the intensities) 
the percentage of judgments "trochaic" also increases, the objective 
intervals remaining equal. In other words, to destroy a trochaic 



20 A QUANTITATIVE STUDY OF RHYTHM 

group which is strongly enough trochaic to be judged trochaic in 
95 per cent, of the trials, a larger increase in the interval after the 
emphasized sound is necessary than that required to destroy a 
trochaic group which is only strong enough to be judged trochaic 
in 60 per cent, of the trials. We may conclude, then, that the 
magnitude of A-B, when positive, may be taken as a measure of 
the degree of trochaism, when negative, of the degree of iambism, 
in so far as this trochaism or iambism is due to other factors than 
objective differences in the intervals between the stimuli. 

No use has as yet ever been made of this rhythm indifference 
point. McDougall, as I have already explained, made use of the 
indifference point in the estimation of time intervals. The ques- 
tion asked of his subjects "was invariably as to the apparent rela- 
tive duration of the two intervals,"^ and the indifference point at 
which he arrived represents "the quantitative proportion of the 
two durations necessary to produce the impression of temporal 
uniformity in the series."^ The subjects who took part in the 
present investigation, however, were instructed, except in certain 
cases mentioned later, first, to judge whether the sounds produced 
an impression which they would speak of as rhythmical, and second, 
in case there was rhythm, to indicate as best they could the nature 
of the rhythm. In their judgment as to the quality of the rhythm, 
they usually made use of the terms iambic, trochaic, etc.; but as 
full an introspection as the subject was able to make was taken on 
every rhythm to which he listened. Nothing was said to the subjects 
about duration of intervals or temporal uniformity. The indiffer- 
ence point here means a point at which the rhythm is no more one 
type than another. The iambic-trochaic indifference point is the 
point at which the impression is no more trochaic than iambic, but 
at which a slight increase in the interval following the accented 
sound causes the impression to become one of iambic rhythm, and a 
slight decrease in the same interval changes the impression to that 
of trochaic rhythm. That such an indifference point is also the 
indifference point for the perception of the time intervals is by no 
means self-evident, and the question can be settled only by experi- 
ment. I shall show^ that the two indifference points correspond 
very closely indeed, but are not quite identical. 

I have defined the indifference point as regards rhythm as that 
point at which the impression is no more that of one rhythm than of 
another, ^. e., the point at which one rhythm is just as natural and 

* Monog. Sup. Psychol. Rev., 4, 379, 1903. 
^ Ibid., p. 378. 
3 Chap. VI. 



APPARATUS AND PROCEDURE 21 

just as easy as another, instead of defining it as the point where no 
rhythm at all exists, because rhythm may occur even at the indiffer- 
ence point. As Professor Woodworth has pointed out, the same 
series may be heard in different rhythms.^ The following intro- 
spections from subjects who have had a great deal of experience 
with rhythms near the indifference point are very definite. Subject 
Ws. writes: "At the very point where iambic turns trochaic, there 
seems to be no rhythm at all. The reason I put down 'doubtful' 
is because I can determine no rhythm. If I could determine rhythm, 
I could tell whether it was iambic or trochaic. At the turning point, 
there is just a series of sounds, one louder than the other. It seems 
to me that at one point there is formed a continuous, even, un- 
divided series, and no rhythm exists. At this point, you can make 
the rhythm either iambic or trochaic, in your mind." Subject Ww. 
says: "My general conclusion concerning the indifference point is 
that I can get any one of three things — no rhythm at all, iambic, 
or trochaic — any of the three cases may occur depending upon the 
way I attend to the sounds— the way I listen to them." Subject 
Br, who always counted when he obtained the rhythmical effect, 
except in cases where he purposely avoided it, gives the following 
introspection: "At the time when the notes and intervals were all 
equal, the idea of number, that is, the impulse to count one, two, 
dropped out. An idea of mere succession remained." These 
introspections seem sufficient to establish the fact, that, at the in- 
difference point, rhythm may entirely disappear, but also that 
either one of two rhythms may be obtained with about equal facility. 
Notwithstanding the fact that the impression of rhythm produced 
by a series of sounds depends to a large extent upon subjective 
factors, it is none the less true that it depends largely on the nature 
of the sound series, and it is by no means impossible to study the 
relation between the series of stimuli and the ensuing impression. 
When the series of sounds is not near the indifference point, the 
rhythm perceived by any one subject as the result of any given 
series of stimuli is practically always the same. And if the subjects 
are instructed in all cases to indicate which rhythm is the most 
natural or the easiest, it will be found that the indifference point is 
really a quite narrow 7one, though of course a variable quantity. 
I instructed the subjects in this manner; and throughout the follow- 
ing work, whenever it is indicated that a certain series of sounds 
produced a certain impression of rhythm, it is not meant that any 
other rhythm was absolutely impossible, but that the subject found 
the rhythm indicated to be the most natural. To indicate the 
' /. of Phil. Psych, and Sci. Meth., 4, 17, 1907. 



22 A QUANTITATIVE STUDY OF RHYTHM 

manner in which the instructions were followed I cite the following 
introspection from Ws: "Each time, before determining whether 
the rhythm is iambic or trochaic, I try it each way. When the 
loudest and longest note comes first, I put down trochaic; but when 
it comes at the end of each rhythm, I put down iambic. When 
I can not distinguish any definite rhythm or tell which comes first, 
the loud or the soft tone, I put down uncertain. It is not like an 
impression which comes immediately but which comes after listening 
a while. I do not notice the length of intervals especially." Sub- 
ject Wr writes that "Possibly the judgment iambic or trochaic does 
not mean always that this is obtained the easiest, using easiest in 
the sense of least effort, but perhaps it has occasionally meant 
merely most satisfying." All subjects reported that there was no 
difficulty in following the instructions. Whenever they were not 
sure of the rhythm, they indicated the fact. 

The method used to arrive at the indifference point was the method 
of minimal changes. The subject was first presented a series of 
sounds concerning the rhythmical nature of which there was no 
doubt, a series concerning which he could judge "plainly iambic," 
or, "plainly trochaic," etc. The time allowed for this judgment 
in the early part of the investigation was 45 seconds, i. e., the rhythm 
was allowed to run along unchanged for 45 seconds. In the greater 
part of the work, however, the rhythm was continued until the sub- 
ject made up his mind, v^^hich event was indicated to the operator 
by an electric bell signal. In case the subject had not made up 
his mind, however, at the end of one minute, the rhythm was stopped 
anyway, and the subject, in these cases, wrote "doubtful." If the 
judgment "doubtful" occurred on the first rhythm of the series, 
this judgment was thrown out and the series commenced at some 
other point where the judgment was that the rhythm was plainly 
one thing or another. The theory of the method of minimal changes 
seems to require that we take as our starting-point some point 
where the judgment is not in doubt. The judgment "doubtful" 
on the first member of the series occurred very seldom, not over 
twenty times in the whole investigation; and most of these cases 
occurred in the presentation of the first series or two of a new rhythm, 
before the operator had any definite idea as to where the indifference 
point was located, or, consequently, where best to start the series. 
The very few remaining judgments of doubtful on the first member 
of the series occurred in cases where the series was started nearer 
than usual to the indifference point, either in the hope of hurrying 
the progress of the investigation or of preventing fatigue on the part 
of the subject. 



APPARATUS AND PROCEDURE 23 

In case the first judgment was "trochaic," the interval following 
the more intense or the longer sound was slightly lengthened, and 
that preceding shortened by the same amount. This second rhythm 
was then given for 45 seconds, or until the subject had made up his 
mind. This procedure was continued until the subject judged 
"plainly iambic." This whole series was then repeated many times 
either in the same or reverse direction. In case the first judgment 
was "iambic," then naturally the succeeding rhythms of that series 
were produced by shortening the interval following the louder or 
longer sound and lengthening the interval following the weaker 
or shorter sound. Similarly, to pass from dactylic rhythm to ana- 
paestic, the interval following the emphasized sound was increased 
while the others were decreased. The number of rhythms, i. e., 
the number of steps, in any one series, varied from four to fifteen, 
but was usually between five and nine. The subjects knew that 
the change from one rhythm to another was effected by changing 
the intervals, and that the direction of this change was such that 
the rhythm would finally go over to something else. They were 
ignorant of the starting point and the size of the steps. They were 
informed that the size of the steps might vary considerably or might 
not vary at all. As a matter of fact, their size varied in different 
series from .02 to .06 second, in the case of rhythmical measures 
the total duration of which was 1.5 seconds, and correspondingly 
for longer or shorter groups. This means that in rhythmical groups 
the total duration of which was 1.5 seconds the same interval in 
any two successively presented rhythms of the same series would 
vary only by from .01 to .03 second, inasmuch as one interval was 
lengthened as the other was shortened. The most usual variation 
between any two consecutive steps was ,016 second. Very often 
the subject could notice absolutely no difference in two successive 
rhythms. I believe that, under the conditions of this investigation, 
the size of the steps was usually about what would correspond to a 
just noticeable difference. For any one series of minimal changes, 
the steps were the same throughout. 

It is quite conceivable that under some conditions the judgment 
concerning rhythm might be strongly influenced by which sound 
was heard first. There exists, e. g., a tendency, under certain 
conditions, for persons to hear a series of sounds as trochaic in case 
the louder note first reaches the ear, though the same series is judged 
iambic in case the weaker note is the first to reach the ear, or, as. 
I believe may be said with greater truth, the first to receive attention. 
There can be no doubt of this tendency. I remember that, at the 
beginning of this investigation, I was rather surprised to hear a. 



24 



A QUANTITATIVE STUDY OF RHYTHM 



subject judge a rhythm to be trochaic which I had judged to be 
plainly iambic; but, on listening a second time to the same series, 
I, too, got the impression of trochaic rhythm; and I soon found that 
which I got depended on which note I heard first. I think it probable 
that this effect is due to the fact that a certain direction of attention 
is prescribed by the way the rhythm starts out, inasmuch as this 
same shifting of the rhythm may occur within one and the same 
series by shifting the direction of attention, at least according to 
the introspection of all the subjects I have questioned on this point. 
The instructions given to my subjects prevented this influence, of 
the way the rhythm starts out, from affecting my results. Inasmuch 
as the subjects were to judge which rhythm was the more natural, 
in those cases in which there could possibly be any doubt concerning 
the rhythm, and, with some subjects, in every case (as introspec- 
tions quoted above show), the subjects tried both rhythms to see 
which was the more natural. Hence it would not matter which 
way the rhythm started. I have proved that this is the case by 
keeping throughout quite a period a record of which sound began 
the rhythmical series. The following table proves that it did not 
matter for the purpose of this investigation which sound began 
the series. The table represents the readings in degrees, on Meu- 
mann's machine, of the movable contact, at the rhythm indifference 
point, first, using series in which the longer sound was given first 
in each case, and second, using series in which the rhythms were 
always begun on the shorter sound. The temporal value of 1 degree 
during this work was .00833 second. N refers to the number of 
series of minimal changes. 



Subject 


I<onger sound 
given 1st. 


N. 


Shorter sound 
given I St. 


N. 


Ws 


102.5 


10 


102.5 


II 




106.8 


9 


106.3 


II 




1 10. 8 


9 


112. I 


16 




122.9 


15 


121. 3 


14 




III. 9 


4 


III. 4 


7 




84.6 


17 


85.0 


16 




101.5 


17 


102.0 


II 




97.1 


6 


95-8 


5 




97.0 


5 


96.0 


4 




73-3 


14 


75-0 


14 




III. 3 


14 


III. 7 


13 




107 -5 


14 


107 -5 


14 


Py 


105.0 


II 


1 14.0 


16 




113. 8 


14 


103.4 


15 




102.0 


6 


102.0 


6 


Wr 


102.5 


10 


102.5 


II 




lOI.O 


8 


103.4 


7 




86.0 


16 


83.1 


6 



APPARATUS AND PROCEDURE 25 

Notwithstanding that the above results show that, under the con- 
ditions of this research, the results were not affected in any very 
appreciable degree by the question as to which sound came first, 
nevertheless the precaution was taken throughout to begin about 
equally often with each note, both in the case of two-membered and 
three-membered rhythms. 

While the method of minimal changes has been used almost 
exclusively in this research, I have also occasionally used a method 
of constant stimuli, and cite below results on the same point by the 
two methods. Both methods, of course, have advantages and 
disadvantages, but for the purpose of this research no other method 
than that of minimal changes could even be seriously considered, 
because of the greater time required, and it would have been ab- 
solutely impossible to get any of the subjects to work another 
hour longer than they did. As it was, three of the subjects, Ws, 
Dy, and Ww, served for over 300 hours each. From the experi- 
ments I performed with the method of constant stimuli, I estimate 
that 50 per cent, more time would have been required to cover 
the same ground; and, as far as I can see, the results, even then, 
would have been less reliable. 

It may be well to point out that, while I have spoken of the 
method used as that of minimal changes, there was no standard 
stimulus with which a changeable one was compared, and that 
consequently the procedure is different from that of the method 
of minimal changes as ordinarily described. The act required 
of the subject was merely one of introspective description of a single 
stimulus, the stimulus consisting of a series of sounds. To a certain 
extent, the judgment was a matter of identification. Further, it 
may be well to repeat in this connection that the same objective 
rhythm continued for about 45 seconds, during which time, of 
course, a large number, say about 30, separate rhythmical groups 
were presented. The judgment of the subject occasionally under- 
went several fluctuations before a final decision was reached. Sev- 
eral different and more or less opposed judgments might be made 
during the continuance of the same rhythmical stimulus; but, 
unless one of these seemed much less forced than the others, such 
rhythms were judged "doubtful." 

The middle point of this doubtful zone is what is given as the in- 
difference point. In most series, the doubtful zone was quite small. 
In many series, it was less than the size of the steps used in changing 
the rhythm. Thus the judgment "iambic" w'ould be given up to 
a certain point and then a sudden change on the very next step to 
"trochaic" would occur. In these cases, the indifference point 



26 A QUANTITATIVE STUDY OF RHYTHM 

is considered as lying halfway between the last "iambic" and the 
first "trochaic" judgment. The treatment of series in which the 
judgments do not follow a regular course has always been a matter 
for difference of opinion. Some authors advocate considering all 
judgments as "doubtful" between those judgments which are 
alike for three successive judgments, while others advocate neglect 
of all return to the judgment "doubtful" or to the judgment which 
began the series after one sure judgment has been made which is 
different from the sure judgment which began the series. Both 
procedures are plainly unjustifiable theoretically. The question 
might possibly be settled on an empirical basis, but never has been. 
The method I have used for treating such series, while one that has 
never been rigidly advocated, I believe to be the only one that 
does not lead to absurdities. The method used consists in shifting 
the doubtful and reverse judgments all to the middle before taking 
the average. In part, this procedure follows conventional usage. 
Thus, the rhythm indifference point in the following series would, 
by most persons, I presume, be considered as lying between numbers 
5 and 6, i. e., half way between numbers 3 and 8. 



Place in series 
I 


Judgment 
iambic 


2 


iambic 


3 


iambic 


4 


trochaic 


5 


iambic 


6 


trochaic 


7 


iambic 


8 


trochaic 


9 


trochaic 


lO 


trochaic 



We should get the same result if we shifted judgments 4, 5, 6, 
and 7 in such a way that all the iambic lay together and all the 
trochaic together, i. e., changed the places of the trochaic judg- 
ments 4 and 6 with the iambic judgments 5 and 7. For the sake 
of consistency, then, any series of judgments should be treated in 
the same manner. The following two cases will serve as illustrations. 

Place in series Judgment 

1 iambic 

2 iambic 

3 iambic 

4 doubtful 

5 iambic 

6 trochaic 

7 trochaic 

8 trochaic 

Indifference point in above series is at 5. 



APPARATUS AND PROCEDURE 27 



e in series 


Judgment 


1 


iambic 


2 


iambic 


3 


iambic 


4 


trochaic 


5 


iambic 


6 


trochaic 


7 


trochaic 


8 


doubtful 


9 


trochaic 


lO 


trochaic 


II 


trochaic 



Indifference point in above series is at 5. 

In order to save time and prevent fatigue, I told all subjects 
to consider the series ended when they had obtained two judgments in 
succession different from the beginning judgment. As a matter of 
fact, the results of this research would hardly be noticeably affected 
by the method used in calculating the indifference point as long as 
any recognized method was followed. But I consider it advisable 
to have some one definite rule at the start which gives a reasonable 
result in any conceivable case. 

I have not considered it worth while to give in the tables which 
follow the range of the zone of doubtful cases. Nor have I tried 
to work out the correlation between the size of any hypothetically 
defined doubtful zone and the mean variation of the indifference 
point in successive series. Inasmuch, however, as neither the 
method of minimal changes nor that of constant stimuli has ever 
before been systematically applied to the direct investigation of 
rhythm, that is, to investigations in which the judgment rendered 
by the subject is a designation of the nature of the rhythm he gets, it 
may be worth while, in order to give some general idea of the dis- 
tribution of judgments, to cite the following records selected as 
typical : 

Method of minimal changes. Subject, Py. Rhythmical stimuli 
consisting of an alternation of two sounds of unequal, but constant, 
length, and of equal intensity. Longer sound = .09 second, 
shorter = .06 second. In the column headed "after," is indicated 
the duration of the interval after the longer sound; in that headed 
"before," the duration of the interval before the longer sound. 

The table below represents the results for only one series of mini- 
mal changes. The total number of series of minimal changes used 
in this investigation was about 24,000. The indifference point in 
this series is taken as the point at which the interval after the 
longer sound is .53 second. 



28 



A QUANTITATIVE STUDY OF RHYTHM 





Duration 


of 


inteiA'als 


before and 


after 


Subject's 




Ion 


gcr sound, in 


seconds 




judgments 




AiUr. 








£jfore. 






G/ 









.3 9 


iambic 




.59 









.4? 


iambic 




.5 7 
.55 









.43 
.■45- 


iambic 

iambic 




.53 









.47 


• staccato 




.51 
.49 









.4-9 
.SI 


trochaic 

' trochaic 




.47 






— 


.53 


trochaic 



The distribution of judgments, as well as the relation between 
the methods of minimal changes and constant stimuli, is indicated 
in the following results from subject Ws. The rhj-thmical stimulus 
consisted of an alternation of a loud with a soft sound. Loud soimd 
audible to a distance of 136 feet, soft sound audible to a distance of 
24 feet. Both sounds 0.13 second in duration. Total duration 
of one measm'e, that is, from the beginning of one loud sound to 
the beginning of the follomng loud sound = 1.5 seconds. The 
method of constant stimuli used consisted merely in giving the same 
objective rhythms as in the method of minimal changes, but in 
irregular order. 

Method of constant stimuli. 



Intervals before and after 
louder sound, in seconds 



Before 


Aftei 




66 

62 


•57 

.62 




59 
57 
55 


.64 
.66 
.68 




53 


.70 




49 


•74 



Subject's judgments (percent) 



Trochaic 


Doubtful 


Iambic 


N 


100. 


0.0 


0.0 


26 


85.7 


14-3 


0.0 


42 


71-4 


2S.6 


0.0 


56 


31-3 


59-4 


9-3 


64 


0.0 


16.0 


S4.0 


50 


0.0 


5-5 


94-5 


54 


0.0 


0.0 


100. 


36 



Judging from the above table, the rhythm indifference point 
is about at the point where the interval before the louder sound is 
.57 second and that follo-^ing it .66 second. It ■v\-ill be observed, 
also, that a change by one step from this point is sufficient to make 
the judgment in over 70 per cent, of the cases either iambic or 
trochaic depending upon the direction of the change. Moreover, 
in this case, as in others with the same and other subjects, the results 
obtained by this method agree very closeh' with those obtained 
by the method of minimal changes. Thus, in the present instance, 



APPARATUS AND PROCEDURE 29 

the indifference point for the same rhythmical series of sounds 
on the same morning, by the same subject, was found, as the result 
of ten series of minimal changes, to be at that setting which gives 
as the interval before the louder sound .57 second (absolute M. V. = 
.01 second) and that following it .66 second (absolute M. V. = .01 
second) — the same result as with the method of constant stimuli. 



CHAPTER III 

Intensity 

The method used for the investigation of the effect of intensity 
in rh3^thm consists in finding the indifference point for rhythm with 
different ratios of intensity between the sounds. The method of 
determining the indifference point and the method of measuring 
the intensity of the sounds have been described in Chapter II. 
Rhythms in which every alternate sound is objectively accented 
were the ones given most attention. At the rate at which these 
were usually given, one sound every .75 second, all the subjects 
invariably got a two-group. Enough work was done on three- 
membered groups, however, to show that the same general laws 
hold in both cases. 

In the experiments represented in table I., the duration of all 
the sounds was kept constant and was .13 second, while the total 
duration of one measure, that is, the time from the beginning of 
one louder sound to the beginning of the following louder sound, 
was also kept constant at 1.5 seconds. There were just two different 
intensities of sounds used in any one rhythmical series, every second 
sound being louder; and all the louder were of one intensity, and all 
the weaker of the other intensity. The intensity of the sounds 
is indicated by the distance, in feet, to which they were just audible. 
For all the tables presented in this chapter, the weaker sound was 
just audible at 24 feet. In the column headed A-B, is indicated 
the duration, at the indifference point for rhythm, of the interval 
following the louder sound minus the duration of the interval preced- 
ing the louder sound. By "before" is meant the interval of silence 
before the louder sound, i. e., the time from the end of the preceding 
sound to the beginning of the louder sound; by " after" is meant the 
interval of silence following the louder sound, ^. e., the time from 
the end of the louder sound to the beginning of the following sound. 
The quantity A-B, therefore, is obtained by subtracting the cor- 
responding quantity in the column "before" from that in the column 
"after." N refers to the number of series of minimal changes run 
through in the determination of any one indifference point. The 
mean variation (MV) is the mean variation in the duration of the 
intervals at the indifference point. All the results obtained with 
any one ratio of intensities are treated as one group in obtaining 
the mean variation. Inasmuch as one interval was lengthened as 
the other was shortened, and vice versa, the absolute M. V. of one in- 
terval is also that of the other; and for that reason, I have given 



INTENSITY 



31 



the absolute M. V. rather than the relative. By "louder sound/' 
is meant the distance to which the louder sound was just audible. 
All durations are given in seconds. 



Subject 
Wr 



Dy 



Ww 



Ws 



TAB 

tive Intensity on the 


LE I 
Iambic-trochaic 


Indifference 


Point. 


ttd just audible at 24 feet. 








der sound Before After 


N 


My 


A — B 


30 


613 


617 


10 


.008 


.004 


32 


603 


627 


9 


.005 


.024 


40 


576 


654 


9 


.008 


.078 


70 


567 


663 


31 


.025 


.096 


136 


555 


675 


10 


.010 


.120 


196 


548 


682 


15 


.016 


•134 


300 


547 


683 


10 


.012 


.136 


420 


536 


694 


25 


.024 


.158 


616 


513 


717 


II 


.013 


.204 


IIOO 


509 


721 


17 


.016 


.212 


28 


592 


638 


6 


.012 


.046 


30 


573 


657 


II 


.012 


.084 


32 


564 


666 


II 


.014 


.102 


40 


532 


698 


18 


.020 


.166 


70 


532 


698 


33 


.023 


.166 


136 


506 


724 


15 


.016 


.218 


196 


503 


727 


23 


.024 


.224 


300 


499 


731 


28 


.023 


•232 


420 


503 


727 


10 


.014 


.224 


616 


484 


746 


22 


.oig 


.262 


800 


456 


774 


15 


.022 


.318 


IIOO 


457 


773 


34 


.024 


.316 


28 


594 


636 


22 


.022 


.042 


30 


581 


648 


28 


.016 


.068 


32 


573 


657 


20 


.oig 


.084 


40 


568 


662 


19 


.026 


.094 


70 


558 


672 


23 


.024 


.114 


136 


554 


676 


25 


.015 


.122 


196 


546 


684 


21 


.022 


.138 


300 


546 


684 


21 


.020 


.138 


616 


534 


696 


21 


.028 


.162 


IIOO 


534 


696 


20 


.016 


.162 


28 ■ 


614 


616 


20 


.022 


.002 


30 


600 


630 


20 


.oog 


.030 


32 


590 


640 


20 


.012 


.050 


40 


582 


648 


38 


.019 


.066 


70 


577 


653 


20 


.021 


.076 


136 


568 


662 


58 


■033 


.094 


196 


568 


662 


56 


.025 


.094 


300 


566 


664 


41 


.021 


.098 


420 


559 


671 


44 


.028 


.112 


616 


558 


672 


46 


.028 


.114 


800 


563 


667 


35 


.oig 


.104 


IIOO 


550 


680 


38 


.018 


.130 



32 A QUANTITATIVE STUDY OF RHYTHM 

From the preceding table, it is obvious that with equal intervals 
and every second sound accented the rhythm is trochaic, that is, 
the effect of accent is to cause the accented sound to appe.ar grouped 
with the following weaker sound. In chapter II it was shown 
that when the quantity A-B is positive, its magnitude measures 
the trochaism of the sound series. It is obvious, therefore, from 
Table I, that the rhythm becomes more and more trochaic as the 
ratio between the intensities of the louder and weaker sounds in- 
creases. It is further obvious that the trochaism of the series in- 
creases at first very rapidly and later very slowly relative to the 
increase in distance to which the louder sound is just audible, the 
weaker remaining constant. The same statement can be made 
concerning the increase in the intensity of a sensation as the energy 
of the stimulus increases. It is therefore not impossible that the 
increase in trochaism tends to be proportional to the increase in 
the ratio of the sensation intensities of the sounds, but not enough 
is known of the relation between the energy of sound stimuli and 
the intensity of sensations to make speculation on this point profit- 
able. 

That the effect of an objective accent in three-membered groups 
is similar to that in two-membered groups is shown by Table II, 
which presents results obtained when every third sound was the 
more intense. The indifference point for these tables is the dactylic- 
anapaestic, and was obtained in the same way as in the case of 
groups of two sounds each, namely, by shortening the interval 
preceding the more intense sound and at the same time lengthening 
the interval following it, or vice versa. The interval between the 
two weaker sounds, which were both of the same intensity, (24 feet), 
was kept constant throughout, and equal to .577 second. As 
before, the quantities in the column headed A-B represent the 
duration of the interval following the louder sound minus the dura- 
tion of the interval preceding it. The duration of all the sounds 
was .09 second, and the total duration of one measure, that is, 
the time from the beginning of one louder sound to the beginning 
of the succeeding louder sound, was 2.0 seconds. 

TABLE II 
Effect of Relative Intensity on the Anapaestic-dactylic Indifference Point. 
Interval between the two weaker sounds (constant) = 0.577 seconds. 
Weaker sound just audible at 24 feet. 



ibject 


Louder sound 


Before 


After 


N 


M V 


A — B 


Dy 


32 


•541 


.613 


ID 


.028 


.072 




40 


•525 


.629 


10 


.022 


.104 




136 


•517 


•637 


9 


.013 


.120 




616 


•479 


.675 


ID 


.020 


.196 







INTENSITY 








abject 


I,ouder sound 


Before 


After 


N 


MV 


A — B 


Ww 


32 


•540 


.614 


10 


.016 


.074 




40 


■523 


.631 


8 


.013 


.108 




136 


•499 


.655 


12 


.016 


.156 




616 


•475 


.679 


15 


.017 


-204 



33 



This same law, namely, that the accented stimulus tends more 
and more strongly to begin the group with increase in the ratio of 
intensities of the stimuli, was found to hold good of rhythms in which 
the stimuli were electrical shocks instead of sounds. The same 
apparatus was used as in the investigation of sound rhythms except 
that in place of the telephone, which had produced the sounds, 
was substituted a pair of sponge electrodes, which were tied to 
motor points of the arms. Subject Dy gave the introspection that 
the impression of rhythm was "fully as strong in the case of elec- 
trical stimulation as in the case of sound." My own introspection 
is that there is about as much rhythm in one case as the other. 
Every third stimulus was the stronger and consequently each 
group consisted of three movement percepts (each shock producing 
a twitch of the muscles of the arm): and the indifference point 
obtained was the dactylic-anapaestic. The results for electrical 
stimulation are presented in Table III. The total duration of one 
measure is 2.0 seconds, and the duration of each stimulus equals 
.09 second. The interval between the two weaker shocks, which 
were both equal in intensity, was kept constant throughout and was 
equal to .577 second. 

TABLE III 

Effect of Relative Intensity on Anapaestic-dactylic Indifference Point when 

Electric Shocks Are Used as Stimuli. 

Interval between the two weakest shocks (constant) = 0.577 second. 
Subject stronger stimulus Before After N MV A — B 



Dy weak 

medium 
very great 

Ww weak 

medium 
very great 



566 
553 
513 

567 
561 
533 



588 6 .022 .022 

601 8 .017 .048 

641 8 .027 .128 

587 6 .014 .020 

593 4 -016 .032 

621 6 .018 .088 



In the work on sound, all subjects frequently remarked on the 
greater apparent duration of the louder sound. The difference both 
in loudness and in duration seemed greater when the rhythm was 
pronounced than when near the indifference point. 



CHAPTER IV 

Rate and Intensity 

It was desired to study here the effect on the rhythmical impres- 
sion of changes in the rapidity with which the series was run off. 
These changes are those produced by varying the rate of rotation 
of the Meumann's time-sense apparatus. The absolute duration of 
the measure and of all its parts, intervals and sounds, was varied, 
while all relative durations were kept constant. One method of 
changing the rate of a series of sounds is to change only the intervals 
of silence between the sounds. Such a procedure introduces changes 
in a factor other than rate, namely, the proportion between the 
duration of the sounds and the duration of the intervals. In the 
present experiments, the absolute duration of the sounds was varied 
in direct proportion to the absolute duration of the intervals between 
the sounds, so that the proportion of sound to silence and the pro- 
portion of either sound or silence to the whole measure was kept 
constant. The intensities of the sounds were also kept constant. 

Tables IV to VI show the effect of variation in the rate of rhythmi- 
cal series in which every second sound is the louder. The subjects 
perceived the sounds in groups of two except for the measures whose 
total duration is indicated as .75 second or .5 second. At these 
last mentioned rates, the sounds were grouped by four, so that as 
a matter of fact at these two rates the total duration of the groups 
was twice that indicated in the tables as the duration of the measure; 
but the time from the beginning of one of the loud sounds to the 
beginning of the following is indicated as the total duration of the 
measure, at these rates as in the other cases, for the sake of readier 
comparison. In the following tables are presented the duration 
of the intervals at the indifference point. In the column headed 
"measure" is indicated the duration of the cycle from the beginning, 
say, of one loud sound to the beginning of the following loud sound. 
The expressions, N, M V, A-B, before and after, have the same 
meaning as in the previous chapter. The quantity A-B, however, 
which represents the amount by which the interval after the accented 
sound is longer than the interval before it, at the indifference point 
for rhythm, is expressed in the following tables as a percentage 
of the duration, at the indifference point, of the interval before 
the accented sound. All durations are given in seconds. 



RATE AND INTENSITY 35 

TABLE IV 

The Effect of Rate on the Iambic-trochaic Indifference Point. 

Louder sound just audible at 70 feet; weaker, at 24 feet. 
Duration of every sound relative to the total duration of one measure = 
13 to 150. 



ubject 


Measure 


Before 


After 


N 


M V 


A — B 


Wr 


I.O 


•38 


•44 


13 


.008 


15.8 




1-5 


•57 


.66 


14 


.014 


15-8 




2.0 


•77 


.87 


14 


.007 


13.0 




30 


1. 14 


1.32 


19 


■054 


15.8 




4.0 


1-52 


1.76 


14 


.027 


15.8 




5-5 


2. 12 


2-39 


14 


■055 


12.7 




7-5 


3-05 


3-II 


17 


.100 


2.0 


Ws 


1.0 


•39 


•44 


II 


.006 


12.8 




1-5 


•58 


• 65 


II 


.011 


12. 1 




2.0 


■78 


.87 


ID 


.016 


II-5 




■ 2.5 


•97 


1 .09 


8 


.026 


12.4 


Dy 


0.5 


.18 


•23 


18 


.007 


27.8 




0.75 


.29 


•34 


19 


.008 


17.2 




1-5 


•57 


.66 


II 


.021 


15.8 




2.0 


•77 


•87 


15 


.015 


13-0 




3-0 


1^15 


131 


10 


.032 


13.9 




4.0 


1^55 


1^73 


10 


.040 


11.6 




5-5 


2.24 


2.27 


9 


.051 


1.3 




7.0 


3.08 


3.08 


8 


.og6 


0.0 



TABLE V 

The Effect of Rate on the Iambic-trochaic Indifference Point. 

Subject, Ww. 

Louder sound just audible at 136 feet; weaker, at 24 feet. 
Duration of every sound relative to the total duration of one measure 
13 to 150. 



Measure 


Before 


After 


N 


M V 


A — B 


0.5 




16 


•25 


12 


.005 


56.3 


0.75 




26 


•36 


ID 


.010 


38.5 


1 .0 




37 


.46 


ID 


.006 


24-3 


1^5 




53 


.70 


8 


.oop 


32.1 


2.0 




•73 


•91 


II 


.018 


24.7 


2.5 




90 


1. 16 


ID 


.022 


28.9 


3^o 


I 


06 


1 .40 


12 


.oog 


32.1 


3-5 


I 


•29 


1.58 


10 


.021 


22.5 


5-0 


I 


•91 


2. 19 


14 


.041 


14.7 


7-5 


2 


•97 


3^i8 


6 


.104 


7-1 


10. 


4 


03 


4.17 


7 


.083 


3.5 



36 A QUANTITATIVE STUDY OF RHYTHM 

TABLE VI 

The Effect of Rate on the Iambic-trochaic Indifference Point. 
Subject, Dy. 

Louder sound just audible at 136 feet; weaker, at 24 feet. 
Duration of louder sound relative to the total duration of one measure 

37 to 150; of weaker = 13 to 150. 



Measure 


Before 


After 


N 


M V 


A— B 


1 .0 


.29 


•38 


16 


.016 


31.0 


1-5 




44 


•56 


16 


.013 


27.3 


2.0 




59 


•71 


16 


.017 


20.3 


2.5 




75 , 


.92 


18 


.040 


22.7 


3-5 


I 


05 


1.28 


16 


.047 


21.9 


5-0 


I 


47 


1.87 


13 


.072 


27.2 


7-5 


2 


30 


2.70 


16 


■054 


17.4 


10. 


3 


20 


3-47 


16 


.106 


8.4 



In considering the above tables, it is necessary to bear in mind 
the significance of the quantities in the columns headed A-B. As 
I have previously stated, this quantity is a measure of the amount 
of temporal segregation presented by the group, and accordingly an 
index to the amount of rhythm got out of the sound series by the 
subject. We may therefore describe the effect of variations in rate 
as follows. Within certain limits the degree of temporal segregation 
presented by the groups remains very nearly constant. With 
slower rates, the rhythm entirely or almost entirely disappears. 
The limits within which the degree of temporal segregation remains 
relatively constant vary with individuals and with the nature of 
the objective series. Thus, in the case of subject Dy, in the case 
presented in Table IV, where there was merely a difference in in- 
tensity between the sounds, the rhythm dropped off very suddenly 
between a rate which gave as the total duration of one measure 
4.0 seconds and one which gave as the total duration of one measure 
5.5 seconds. With a different sound series, however, one in which 
the louder sound was also the longer (Table VI), a certain amount 
of rhythm persisted even at a rate which gave as the total duration 
of one measure 10.0 seconds; though even in this case there was 
already a very considerable decrease at the rate of 7.5 seconds for 
one measure. In general, the tables indicate that, from a rate of 
1.0 second for one measure up to a rate of between 4.0 and 7.0 
seconds for one measure, the degree of rhythm remains about con- 
stant. 

Sufficient observations have not been made with very fast rates 
to generalize concerning the effect of increasing the rate beyond 1.0 
second for one measure. There is some indication, however, that 
with rates faster than 1.0 second for one measure there is a marked 



RATE AND INTENSITY 37 

increase in the amount of temporal segregation presented by the 
rhythm. As already stated, grouping by four instead of by two 
comes in at rates faster than 1.0 second for one measure. Inasmuch 
as subjective rhythm comes in very markedly at rates of about 
1 second for two beats or faster, the increase above noted in the 
rhythmical effect for the faster rates can hardly be regarded as the 
effect of an increase in rate on the rhythmical effect of intensity or 
duration. It seems rather that we have in the case of these rates 
a subjective factor which is niore or less independent of any differ- 
ences which may prevail in the intensity or duration of the sounds 
composing the series, since at these rates grouping occurs when all 
the sounds are equal both in intensity and duration.^ 

The rate at which intensity and duration completely fail to exert 
any influence favoring rhythm is indicated as being somewhat slower 
than that usually given as the limit of rhythm, being in some cases 
beyond 10.0 seconds for one measure. While a marked decrease in 
the amount of rhythm got from a series occurs in the neighborhood 
of from 4.0 to 7.0 seconds for one measure, the limit at which varia- 
tions in intensity and duration fail to exert any influence making 
for temporal segregation is indicated as varying from 7.0 to over 
10.0 seconds for one measure. Subject Wr, an experienced in- 
trospectionist, told me he got rhythm (when the intervals were not 
too nearly equal) from a rate which made the total duration of one 
measure 26.0 seconds. What he meant was that when the intervals 
were not too nearly equal the sounds seemed grouped by twos, and 
one sound was louder than the other. Now, grouping is usually 
held to be one of the chief characteristics of rhythm. Yet Table IV 
shows for this same subject that, so far as the effect of intensity 
goes, making one sound stimulus about eight times as energetic as 
the other failed, practically, to exert any grouping effect when 
the rate was slow enough to make the total duration of one measure 
equal to 7.5 seconds. It is evidently impossible to fix the upper 
(slow) limit of rhythm until we have a definition of rhythm, but 
by the method I have used it is possible to fix the upper limit at 
which intensity or duration exert any rhythmical effect in the sense 
of promoting temporal segregation. 

All subjects found the work on the very slow rhythms extremely 
fatiguing, and it was hardly possible at the slowest rates to run 
through more than 3 or 4 series of minimal changes at one sitting. 
Extremely close attention is required at these slow rates to get any 
rhythm at all. 

^ Meumann, Phil. Stud., lo, 302, 1894. 



CHAPTER V 

Duration 

In the investigation of the effect on rhythm of variations in 
the duration of the sounds, I did not find so much agreement between 
different individuals as in the case of variations in intensity. In- 
dividual differences, while quite marked in the case of duration, 
are yet not so great as to prevent certain generalizations. 

The investigation of the effect of duration in rhythm was carried 
on in the same way as the investigation of the effect of intensity, 
except that instead of using sounds of different intensity, the in- 
tensity of all sounds was kept constant and their duration alone 
varied. As has already been shown, in the case of two-membered 
rhythms, if we start out with a trochaic rhythm, in which the louder 
sound begins the measure, by gradually increasing the interval 
after the louder sound, we arrive at an iambic rhythm, in which 
the louder sound ends the measure. I found no exception to the 
possibility of changing in this manner from trochaic to iambic. 
Similarly, in the case of rhythm in which we have differences in dura- 
tion of the sounds but not in intensity, if we start with a trochaic 
rhythm with the longer sound beginning the measure, by increasing 
the interval after the longer sound and decreasing the interval 
before it, we will arrive at an iambic rhythm, with the longer sound 
second. In this case, the longer sound ordinarily seems to the 
subject to be accented. I found no exception to the possibility 
of changing in this manner from trochaic to iambic in the case of 
any of the subjects that I have worked on, and I have tested thirteen 
subjects in this way. In the case of one subject however, subject 
Sh, the 7nost usual result was not a change from trochaic to iambic. 
With a sufficient increase in the interval following the longer sound, 
the rhythm, instead of changing to iambic, remained trochaic, but 
a trochaic in which the shorter sound was described as accented. 
In other words, in the case of this subject the first sound of the group 
was usually accented, irrespective of whether the first sound was 
the longer or the shorter. While I did not get this result in any 
other case, three other subjects occasionally stated that they had 
a tendency to hear a trochaic with the short sound accented instead 
of an iambic with the long sound accented, when the interval after 
the longer sound appeared the longer. When the interval before 
the longer sound appeared greater than that following it, the longer 
sound always seemed accented, that is, in no case was any tendency 



DURATION 39 

found to accent the second sound of a two-membered group, when 
the second sound was the shorter and of equal intensity with the 
first. 

This result seems to indicate that there are two separable ob- 
jective factors tending to produce subjective accent, at least in 
some subjects. There is a tendency to accent the longer sound 
and also a tendency to accent the sound which seems to begin the 
group. In some subjects there is no evidence of the latter tendency, 
and only in one subject out of thirteen, subject Sh, was it as strong 
as the tendency subjectively to accent the longer sound. Only 
on the supposition of these two separable tendencies does it seem 
possible to explain the fact, that, when the longer sound appears 
to begin the group, it invariably receives an accent, whereas when 
the longer sound ends the group, it occasionally appears less ac- 
cented than the shorter sound. In the one case the two factors 
work together; in the other, in opposition. 

The apparent intensity of stimuli which, objectively, differ only 
in duration, is a point of considerable interest. Miner found in 
his investigation of rhythm that an increase in the duration of a 
light was mistaken for an increase in intensity. It is, further, 
a well-known fact, that, in the case of sound, the apparent loudness 
of a sound increases with the increase in duration, the objective 
intensity remaining constant. This increase in apparent loudness 
occurs at first very rapidly and then slower, as the duration is in- 
creased; but the maximum of apparent loudness, for sounds of the 
objective intensity used in this investigation of duration in rhythm, 
would hardly be reached before the duration had reached at least 
one second.^ 

The significance of this fact of increase in apparent intensity 
with increase in duration is very definitely indicated by the follow- 
ing introspection from subject Wr, on a rhythm in which the longer 
sound was .25 second and the shorter .13 second, and the rhythm 
given first with both sounds rather loud and then with both sounds 
rather weak. ''There is no doubt but emphasis goes with length: 
tendency to accentuate short note practically disappeared with 
weaker intensity." This tendency to accentuate the short note 
disappears with weak intensities because, when the sounds are 
weak, a slight difference in duration is of more significance for 
their apparent relative loudness than when both are quite loud. 
The results we have just spoken of above, however, show that the 
greater apparent intensity of one or the other sound is not due 

' Kafka, "Uber das Ansteigen der Tonerregung," Psychol. Stud., 2, 256- 
292, 1907. 



40 A QUANTITATIVE STUDY OF RHYTHM 

merely to the fact that a sound increases in apparent intensity with 
increase in duration. In the case of subject Sh, and others, in which 
the first sound of the group was sometimes accented whether the 
longer or not, we have an indication that some process in the central 
nervous system is exerting an effect on the apparent loudness of 
the sounds. I have also many series of introspections which show 
that the difference in apparent loudness of the two sounds, when 
the apparently louder is second, is greater as the interval before 
the second sound is shortened and the interval before the first 
or apparently weaker is lengthened. The reverse introspection 
was never obtained. That is to say, in an iambic rhythm, with 
the second sound apparently louder than the first, the apparent differ- 
ence in loudness is often reported as decreasing as the intervals become 
more and more equal. Similarly in a trochaic rhythm, the first 
sound often seems to decrease in relative loudness as the intervals 
are made more nearly apparently equal. These introspections were 
given by subjects Br, Ww, Py, Dy and Ws. When the intervals were 
nearly equal, such expressions as "very slight accent," ''emphasis 
exceedingly light," "accent doubtful, depending on will" were 
obtained, while when the intervals were unequal, such introspections 
as "accent very pronounced," "2nd (or 1st) clearly louder" etc., 
were given. Concerning a series in which the sounds were unchanged, 
but the intervals varied in the usual way, subject Ws writes: "The 
notes are more nearly of the same length as you approach the chang- 
ing point ' (indifference point). Subject Br, concerning a rhythm in 
which the longer sound was 0.07 second and the shorter 0.06 second: 
''The longer notes seem more intense and slightly higher than the 
shorter. When the intervals are about equal, the length and in- 
tensity of the notes also seem about equal. According as the at- 
tention seizes on one note or the other, is the position of accent and 
the grouping determined. At the time the intervals are equal, dif- 
ference in length and intensity practically disappears. The difference 
became more pronounced whenever the rhythm was more pro- 
nounced." 

These results show that, in some cases, at least, the greater ap- 
parent intensity of one or other of the sounds is due to some central 
process, a conclusion similar to what seems inevitable in regard to 
the subjective accent in purely subjective rhythm. I think that 
this process is probably that of attention, but I do not care to insist 
on this at present, as it would require too much space to give all 
the grounds for such a belief. It is some process, however, which 
is quite separate from that involved in the increase in apparent 
intensity resulting from an increase in duration. 



DURATION 41 

The general conclusion, then, is that the longer sound should 
naturally appear the louder, as we might say, for sensory reasons, 
apart from more complicated or more central processes. At the 
same time, there is some central process, possibly attention, 
which has some slight effect on apparent intensity. When the longer 
sound appears the louder there is no necessity for considering this 
as due to greater attention directed to the longer. On the contrary, 
the shorter might be receiving the greater amount of attention and 
yet not appear as loud as the longer, the physiological or sensory 
effect exerted by duration on intensity being able to swamp the slight 
effect of the central process on intensity, in case the two processes 
work in opposition. In those cases, however, where the shorter 
sound seems the louder, the central process is evidently having 
a stronger effect than the sensory. It is possible that all the subjects 
have not used the words intensity, and accent, and emphasis in the 
same sense, and that the effects referred to in some cases as differences 
in intensity were differences in some other sort of emphasis; but 
this makes it all the more desirable to distinguish between the un- 
questionable effect on intensity of increase in duration and the 
effect on something referred to as accent or as intensity, exerted 
equally unquestionably by some other apparently more central 
process. 

The results obtained with regard to the indifference point in 
two-membered rhythms produced by alternation of a longer and 
shorter sound of equal objective intensity and of the same quality 
are presented in Tables VII and VIII. By the total duration of a 
measure, or in Table VIII, by "Measure," is meant the time, say, 
from the beginning of one long sound to the beginning of the next 
long sound. By N, is meant the number of series of minimal changes 
run through in the determination of any one indifference point. 
The tables show the length of the intervals at the indifference point 
for rhythm. By the interval "before,'' is meant the duration of 
the interval before the longer sound, i. e., the time elapsing from 
the end of the short sound to the beginning of the long; and by the 
interval 'Rafter," is meant the interval after the longer sound, i. e., 
from the end of the long sound to the beginning of the short. The 
mean variation referred to is the variation in the length of these 
intervals. All the results obtained from any one ratio are treated 
as one group in obtaining the mean variation. Inasmuch as one 
interval was lengthened as the other was shortened, and vice versa, 
the absolute M V of one interval is also that of the other; and for 
that reason, I have given the absolute M V rather than the relative. 
In the column A-B, I have indicated the difference in duration 



42 



A QUANTITATIVE STUDY OF RHYTHM 



between the interval following the longer sound and that preceding 
it. In the column headed (A + L) — (B + S), is indicated the 
difference between the duration of the iDterval of silence after the 
longer sound + that of the longer sound and that of the interval 
of silence before the longer sound + that of the shorter sound. 
In other words, in this column the measure is considered as being 
composed of two intervals each of which extends from the beginning 
of one sound to the beginning of the following sound. The column 
headed (A + L) — (B + S) presents the duration of the interval 
extending from the beginning of the longer sound to the beginning 
of the shorter minus the duration of the interval extending from the 
beginning of the shorter to the beginning of the longer. The following- 
scheme illustrates this method of treating the measure: 



I measure 



B 



Iv 



(A + L) 



(B + S) 



All durations are indicated in seconds. By "longer" or "shorter'" 
is meant the duration of the longer or shorter sound. 

TABLE VII 
The Effect of Relative Duration on the Iambic-trochaic Indifference Points 
Duration of shorter sound (constant) =0.13 second. 
Total duration of measure (constant) = 1.5 seconds. 



Ws 
(July '08) 



Ws 
(July, '07) 



3 


5 

V 


16 


58 


■ 19 


58 


. 22 


58 


•25 


55 


.28 


55 


•31 


55 


■37 


55 


•43 


55 


.48 


54 


.60 


54 


.18 


50 


.28 , 


55 


•34 


54 


•54 


51 



< 


IZ! 




M 
1 


•4 
< 


63 


31 


.014 


+ 


.05 


+ 


08 


60 


58 


.024 ■ 


+ 


.02 


+ 


0& 


57 


26 


.017 


— 


.01 


+ 


09 


57 


28 


.022 


+ 


.02 


+ 


13 


54 


24 


.020 





.01 


+ 


14 


51 


26 


.010 





.04 


+ 


14 


45 


28 


.014 





.10 


+ 


14 


39 


27 


.020 


— 


.16 


+ 


13 


35 


22 


.010 





.19 


+ 


16 


23 


23 


.018 





•31 


+ 


16 


56 


32 


.020 


+ 


.06 


+ 


II 


53 


18 


.016 


— 


.02 


+ 


1 3. 


48 


41 


.030 





.06 


+ 


13 


•31 


22 


.025 


— 


.20 


+ 


21 



DURATION 

TABLE Yll— {Continued) 



43 



V 

iff 

n 

03 


u 

il 

M 
C 

►4 




pq 


u 
< 


^ 


\ 

^ 


fq 
1 
< 


+ 
< 


Vv^w 


16 


57 


64 


15 


.016 


+ .07 


4-. 10 


(July, '08) 


19 


57 


61 


16 


■ 015 


+ .04 


+ .10 




22 


57 


58 


16 


.023 


+ .01 


+ .10 




25 


54 


58 


16 


■ 015 


+ .04 


+ .16 




28 


54 


55 


10 


.014 


+ .01 


+ .16 




31 


53 


54 


10 


.010 


+ .01 


+ ■19 




37 


51 


49 


15 


.016 


— .02 


4-. 22 




43 


47 


47 


10 


■ 015 


.00 


4-. 30 




48 


43 


46 


10 


.024 


+ .03 


+ •38 




60 


40 


37 


16 


.013 


— .03 


+ •44 




72 


36 


29 


10 


.011 


— .07 


+ .52 




84 


33 


20 


10 


.013 


— .13 


+ .58 


Ww 


18 


56 


62 


31 


.025 


+ .06 


+ .11 


(July, '07) 


26 


52 


58 


23 


.036 


+ .06 


+ •19 




34 


50 


52 


28 


.030 


+ .02 


+ ■23 




42 


48 


46 


21 


■034 


— .02 


4- .27 




54 


43 


39 


25 


.020 


— .04 


+ .36 


Dy 


16 


53 ■ 


68 


42 


.040 


4- .15 


4-. 18 




19 


51 


67 


35 


■037 


+ .16 


+ .22 




22 


53 


62 


33 


.020 


+ .09 


4-. 18 




25 


51 


61 


45 


.o2y 


4- .10 


4-. 22 




31 


46 


60 


33 


.026 


+ .14 


+ .32 




37 


43 


57 


37 


.031 


+ .14 


+ .38 




43 


40 


54 


35 


.028 


+ .14 


4-. 44 




48 


39 


50 


28 


.ojg 


+ .11 


+ .46 




60 


36 


41 


25 


.038 


+ .05 


+ •52 




72 


28 


37 


18 


.041 


+ .09 


4-. 68 




84 


24 


29 


15 


.020 


+ .05 


+ .76 


Wr 


16 


61 


60 


15 


.02^ 


— .01 


4- .02 




19 


58 


60 


II 


.Oil 


+ .02 


4- .08 




22 


58 


57 


25 


.020 


— .01 


+ .08 




25 


54 


58 


16 


.018 


+ .04 


+ .16 




28 


55 


54 


16 


.023 


— .01 


+ .14 




31 


53 


53 


17 


.Oil 


.00 


4-. 18 




37 


54 


46 


15 


.014 


— .08 


+ .18 




43 


51 


43 


17 


.023 


— .08 


+ •24 




48 


49 


40 


19 


.014 


— .09 


4-. 26 




60 


44 


33 


1-9 


.024 


— .11 


4-. 36 



44 



A QUANTITATIVE STUDY OF RHYTHM 



TABLE VIII 
The Effect of Relative Duration on the Iambic-trochaic Indifference Point. 



Ws 



Ws 



.64 



Py 1.49 



Vt I. II 



Br I . 50 



• 05 



13 



.08 



h-r 


w 


.06 


24 


.07 


22 


.08 


23 


. 10 


21 


• 15 


22 


. 20 


23 


• 25 


22 


•13 


39 


•15 


39 


.16 


38 


.20 


38 


•23 


38 


• 25 


37 


•29 


38 


■34 


37 


.18 


55 


■34 


48 


•54 


40 


•13 


39 


. 20 


37 


•25 


37 


. 10 


60 


.18 


60 


27 


59 


■35 


57 


•43 


58 


■52 


55 


60 


55 













+ 






















\ 


1 


1 

+ 




< 


Z 


^ 


1 

< 


< 




30 


ID 


.017 


+ 


.06 


+ 


07 


30 


12 


.018 


+ 


.08 


+ 


10 


29 


10 


.012 


+ 


.06 


+ 


09 


27 


10 


.020 


+ 


.06 


+ 


II 


22 


II 


.019 




.00 


+ 


ID 


17 


II 


.011 


— 


.06 


+ 


09 


12 


II 


.018 


— 


.10 


4- 


II 


49 


10 


.017 


+ 


.10 


+ 


13 


47 


10 


.016 


+ 


.08 


+ 


13 


46 


10 


.oig 


+ 


.08 


+ 


14 


44 


II 


.016 


+ 


.06 


+ 


16 


40 


ID 


.015 


+ 


.02 


+ 


15 


39 


10 


■ 015 


+ 


.02 


+ 


17 


34 


12 


.017 


— 


.04 


+ 


15 


31 


ID 


.012 


— 


.06 


+ 


18 


63 


32 


.032 


+ 


.08 


+ 


II 


54 


26 


■039 


+ 


.06 


+ 


29 


43 


32 


.031 


+ 


•03 


+ 


45 


49 


10 


.016 


+ 


.10 


+ 


13 


45 


II 


.010 


4- 


.08 


+ 


18 


39 


ID 


.017 


+ 


.02 


4- 


17 


72 


II 


.030 


+ 


.12 


+ 


14 


64 


10 


.027 


+ 


.04 


4- 


14 


57 


10 


.014 


• — 


.02 


+ 


17 


49 


ID 


.017 


— 


.08 


4- 


19 


40 


7 


.017 


— 


.18 


+ 


17 


35 


8 


.017 


— 


.20 


4- 


24 


27 


8 


.015 


— 


.28 


+ 


24 



As I have already indicated in Chapter II (pages 17-20), the mag- 
nitude of A-B, when positive, may be taken as a measure of the 
degree of trochaism, when negative of the degree of iambism presented 
by the group when the intervals between the sounds are objectively 
equal. A study of the columns headed A-B in the above tables 
shows that without exception the rhythm becomes less trochaic 
or more iambic as the ratio between the longer and shorter sounds 
increases. In some cases, as with subjects Ww, Ws, Wr and Br, 
the change is one from a trochaic rhythm through an indifference 



DURATION 45 

point to a more and more pronounced iambic. In other cases, as 
with subjects Py, Dy and Vt, the change is one from a very decided 
trochaic to a less stable trochaic. In the case of Vt, however, 
there is little doubt but that the rhythm would have become strongly 
iambic with larger ratios between the durations of the sounds. 
In the case of Dy, the change is very slight; but it is still evident 
enough to render it unnecessary to regard this subject as an exception 
to the general rule. The general law indicated, then, regarding the 
effect of the relative duration of the sounds in auditory rhythms, 
is that the rhythm becomes less trochaic or more iambic as the 
ratio between the longer and shorter sounds increases. As I have 
found this law to hold with several different absolute durations of 
the shorter sound, it seems more or less general, though I am unable 
to present data concerning its limits. 

Subject Sh might prove an exception to this generalization. 
I was unable to study the subject thoroughly enough to be sure 
whether he would or not. In his case, I found that, with a ratio 
of two to one and equal intervals, the rhythm was trochaic with 
the longer sound accented. When the interval before the longer 
sound was made considerably shorter than that after it, the rhythm 
usually changed to a trochaic with the shorter sound accented. 
I believe, consequently, that in generalizing the effects of duration 
on rhythm, it is safer to omit the words iambic and trochaic. I 
would therefore state the law of the effect on auditory rhythms, 
composed of groups of two sounds each, exerted by changes in the 
ratio of the durations of the sounds, as follows: With an increase 
in the ratio between the duration of the longer and shorter sounds, 
if the intervals between the sounds are objectively equal, there is 
an increase in the tendency to obtain a rhythm with the longer 
sound at the end of the group or a decrease in the tendency to obtain 
a rhythm with the longer sound at the beginning of the group. 

It is interesting to note that when the intervals are objectively 
equal, the series is heard as trochaic rhythm when the difference in 
the duration of the sounds is sufficiently small. Out of a total 
of thirteen subjects, only two, Ws and Wr, ever showed any ex- 
ception to this statement, and even in these cases the prevailing ten- 
dency with the smaller ratios is towards trochaic. In every case, 
with a ratio of durations of approximately 2 to 1 (25 to 13), with 
equal objective intervals, trochaic rhythm was the result. 

The most strongly trochaic rhythms produced by duration changes 
are those produced by the smallest ratio of durations. It looks 
as though, were the series continued up to a ratio of one to one, 
the series would be more trochaic than ever. This is, however, 



46 A QUANTITATIVE STUDY OF RHYTHM 

not the case. When there is no difference at all in duration (nor 
in accent) the indifference point was found to be almost an3^where, 
that is, all judgments become doubtful. The strongest trochaic 
which it is possible to produce by duration changes alone, therefore, 
is produced when the longer sound is longer, but only very slightly 
so, than the shorter sound. It may be that the trochaic rhythm 
of the smaller ratios was due chiefly to the difference in apparent 
loudness between the longer and shorter sounds. A comparison 
of the effect of slight differences in duration with slight differences 
in intensity lends support to this idea. If we rank the four sub- 
jects used in the investigation of the effect of intensity with respect 
to the magnitude of the quantity A-B for the two small intensity 
ratios of 30 to 24 and 32 to 24 (feet at which audible), and then 
with respect to the same quantity for small ratios of durations, 
we find that the correlation is perfect. In other words, if a slight 
difference in intensity produces a strong trochaic rhythm for any 
given subject, a slight difference in the duration of the sounds does 
also; whereas if a slight difference in intensity produces only a very 
weak trochaic, the same will be true of a slight difference in duration. 
I believe, therefore, that the trochaic rhythm produced by small 
ratios in the durations is due to the different effect as regards in- 
tensity of a long and short sound. Now this difference in apparent 
intensity should increase as the difference in duration increases. 
But after a certain comparatively short duration is reached, the 
increase in apparent intensity is very slow. It seems probable, 
therefore, that, in rhythms produced by short sounds, we have 
to deal with two separate factors, intensity and duration, even though 
the sounds differ physically only in duration. The effect of intensity 
is to make the longer sound begin the measure, an effect which may 
be spoken of as trochaic, whereas the effect of duration, as such, 
is to make the longer sound end the measure, which usually means 
an iambic effect. With a small difference in the absolute durations 
of the longer and shorter sounds, the effect of intensity is compara- 
tively great; but as the longer sound is increased more and more, the 
intensity difference between the two sounds becomes less and less 
relative to the difference in the duration between the sounds. When 
the difference in duration is sufficiently increased, therefore, we 
find that the trochaic effect of intensity tends to disappear because 
of the iambic effect of duration, that is, we get trochaic rhythm 
with small ratios but tend more in the iambic direction with large 
ratios (the absolute duration of the short sound remaining constant). 
This conclusion is further confirmed by the results on absolute 
duration. The difference in intensity between the longer and shorter 



DURATION 47 

sounds for any given ratio of their durations will be smaller, the 
longer the short sound. And my results on the effect of absolute 
duration, presented in Table X,show that, if the ratio is kept constant, 
the rhythm becomes less trochaic or more iambic, as the absolute 
duration of the sounds increases. 

The effect of the difference in intensity produced by a difference 
in duration may also explain in part at least the case of subject 
Dy, in which there was only a slight decrease in the trochaicness 
of the rhythm with an increase in the ratio between the durations. 
For, with Dy, differences in intensity had much more effect than 
with any of the other subjects investigated for the effect of varia- 
tions in intensity. The greatest magnitude obtained for the quantity 
A-B, in the investigation of intensive differences with Dy, was 
50% greater than the greatest magnitude of the same quantity 
for any of the other three subjects. Inasmuch as the effect of 
intensity is trochaic, we should expect to find the tendency towards 
iambic resulting from large ratios between the durations to be less 
in the case of Dy than in the case of the other subjects. 

I have confined myself chiefly to the investigation of two-mem- 
bered groups; but that the above mentioned general law concerning 
the effect of variations in the relative durations of the sounds holds 
for three-membered groups, as well as two-membered groups, is 
indicated by Table IX, where the quantity A-B changes from 
+ .02 to — .06 with increase in the ratio of duration between 
the longer and the two shorter sounds. The method which was 
used here to determine the indifference point was to vary only the 
intervals immediately preceding and following the long sound. 
The two shorter sounds, both of equal duration, were kept the same 
distance apart throughout, being separated by an interval such that 
this interval plus the duration of one of the shorter sounds equals 
one-third the total duration of the measure. By the interval "be- 
fore," is meant the interval before the longer sound; by the interval 
"after," the interval after the longer sound, as in the tables on 
rhythms made up of groups of two sounds. 

TABLE IX 
'The Effect of Relative Duration on the Anapaestic-dactyHc Indifference Point. 

Subject, Ww. 
Duration of the two shorter sounds (constant) = o.o8 second. Total duration 
of measure = 2.0 seconds. Interval between the two shorter sounds 
(constant) = 0.58 second. 
l,onger sound Before After N MV A — B (A + 1,)— (B + S) 

. l5 .54 .56 ID .01 J +.02 +.10 

.30 .50 .44 II .024 — .06 +.16 



48 A QUANTITATIVE STUDY OF RHYTHM 

Before proceeding to the discussion of the effect of variations 
in the absolute duration of the sounds, it is desirable to bring up 
the question of the effect of change in the duration of the intervals 
while the absolute and relative duration of the sounds remains 
unchanged. This is necessary inasmuch as, in the above tables, 
which show that there is a change in the iambic direction as the ratio 
of the durations increases, there occurs simultaneously with the 
increase in the ratio between the sounds a decrease in the total 
amount of silence within the group. The reason for this is that the 
total duration of the group was kept the same, and evidently, then, 
as one of the sounds was increased in duration in order to get bigger 
ratios, the total amount of silence remaining had to be decreased. 
The question might therefore arise as to whether the change noted 
above in the iambic direction was due to the increase in the ratio 
between the durations of the long and short sounds, or whether 
it was due to the decrease in the proportion of the total duration 
of the group occupied by silent intervals. I have not sufficient 
data to state in a definite way what is the effect on rhythm of varia- 
tion in the duration of the intervals, the absolute and relative 
duration of the sounds remaining constant; but I have data to show 
that it is utterly impossible that the changes noted above, from 
trochaic to iambic rhythm with increase in the ratio of the durations 
of the sounds, could be due to the coincident decrease in the pro- 
portion of the total duration of the measure constituted by the 
intervals of silence. In the case of subject Wr, with a ratio of 
durations of 19 to 13, and equal intervals of .59 second, the quantity 
A-B== + .02 second; with the ratio 43 to 13, and equal intervals 
of .47 second, the quantity A-B = — .08 second, and with a 
ratio of 60 to 13, and equal intervals of .39 second, A-B = — .11 
second. That this change of A-B from + to — (i. e., change 
from trochaic to iambic rhythm) is not due to the shortening in the 
intervals is shown conclusively by the following table, where it is 

TABLE X 

Effect of Proportion of Sound to Silence on Iambic-trochaic Indifference Point 

Subject, Wr. 

Duration of shorter sound = .13 second. 



I^onger sound 


Before 


After 


N 


M V 


A— B 


•19 


•58 


.60 


II 


.Oil 


+ .02 


■43 


•51 


■43 


17 


■023 


— .08 


.60 


•44 


•33 


19 


.024 


.11 


■43 


1. 19 


1.25 


14 


■047 


— .06 


.60 


1.07 


1.20 


18 


.025 


— •13 


.72 


.98 


1. 17 


16 


.025 


— .19 



DURATION 49 

seen that the intervals are in every case about twice as great as in 
the case of the ratio 19 to 13, where the intervals were .59 second 
and the rhythm trochaic, and yet the rhythm in these cases, too, 
becomes strongly iambic as the ratio between the sounds increases. 
Evidently the change in A-B from + .02 second to — .19 second, 
as the ratio between the durations of the sounds increases, can not 
be due to a decrease in the intervals of silence, for these have not 
decreased, but nearly doubled. 

As regards the result of variation in the absolute duration of the 
sounds, the following table seems sufficient to establish beyond a 
doubt that in rhythms in which the sounds of a two-membered 
group are of different durations, with a given ratio of these dura- 
tions the rhythm becomes more trochaic or less iambic as the ab- 
solute duration of the sounds decreases, and less trochaic or more 
iambic as their absolute duration increases. The effect of change 
in the absolute duration of the sounds is exceedingly great, so that 
it is impossible to say of any given ratio of durations that it will pro- 
duce a certain rhythm, even in the same subject: the absolute, as 
well as the relative, duration of the sounds must be specified. 

TABLE XI 

Effect of Absolute Duration on the Iambic-trochaic Indifference Point 

Subject, Ww. 

Ratio between durations of long and short sound = 2 to i. 

I^onger sound Before After N MV A — B 

.09 .63 .81 20 .oop + .18 

.25 .54 .58 16 .on + .04 

.45 .41 .40 20 .01^ —.01 

.90 .87 .77 20 .016 .10 

In the above table, it is true that the intervals have not been kept 
constant, but enough has been stated already concerning the effect 
of intervals to show that the changes here shown from a marked 
trochaic to a marked iambic are due to changes in the absolute dura- 
tion of the sounds. 

A question of fundamental importance in the investigation of 
the effect of duration in rhythm concerns the proper method of 
presenting the results. Should the intervals between the sounds 
be treated separately, or should only the temporal distances from 
the beginning of one sound to the beginning of the next be con- 
sidered? At first consideration, this question may seem to have 
little significance, at least in the present investigation, inasmuch 
as the duration of the sounds and intervals have been measured 
separately, and consequently it is possible to present the resiilts 



50 A QUANTITATIVE STUDY OF RHYTHM 

in both ways, and I have so presented them. And it is undoubtedly- 
true that, if all we want is a description of what sort of rhythm is 
heard from any series of sounds, both the length of the sounds and 
the intervals between them, in case intervals exist, should be stated. 
But if we wish to understand or to explain our data, it is of great 
importance to know from what points to measure in presenting the 
results. For instance, should a rhythmical group be considered as 
being made up of "members" which extend, say, from the beginning 
of one sound to the beginning of the following? Dr. Brown, in his 
investigation of spoken verses writes: "On the whole I have decided 
to consider only the beginning of the syllables."^ In the case of 
spoken rhythms or of any other rhythms in which the sounds are 
not of uniform intensity throughout, the question of measurement 
is perhaps more difficult than in the case of sound rhythms such 
as those used in this investigation. While in the present investiga- 
tion the sounds were of uniform intensity throughout their length, 
it is yet possible that what really counted to the listener was neither 
the actual interval between the sounds nor the time elapsing from 
the beginning of one sound to the beginning of the next, but the 
interval between certain points of greatest subjective stress, the 
subjective stress being due, perhaps, to the occurrence, at certain 
points, of greater attention, of motor performances, or to unknown 
factors; and these theoretically possible points of greatest subjective 
stress might conceivably occur almost anywhere. The significance 
which is to be attached to any data conceriiing the objective con- 
ditions of rhythms in which variations in duration occur depends 
upon the mode of tabulating the results. 

I have obtained some data which seem to me to solve this ques- 
tion concerning the proper treatment of results. As I shall show 
in Chapter VI, the temporal spacing of the sounds at the indifference 
point for rhythm very nearly coincides with that spacing at which 
all the intervals appear equal, a fact which justifies the conclusion 
that perception of rhythm and perception of time are closely allied 
processes. This being the case, it is possible to tell whether in rhythm 
the interval between the sounds or the interval from the beginning 
of one to the beginning of the following is the more significant by 
asking the subject, after having determined his indifference point 
for rhythm, to compare intervals, first asking him to compare the 
intervals between the beginnings of the sounds and then asking 
him to compare the silent intervals between the sounds. Of the 
two kinds of interval comparison, that one which gives results cor- 
responding to the judgments on rhythm will be indicated as the 
^ Archiv. of Psychol., No. lo, 28, 1908. 



DURATION 51 

process more nearly allied to the process involved in judging of 
the rhythm; and consequently the more significant mode of tabu- 
lating the results will also be indicated. In other words, if the re- 
sults concerning the indifference point for rhythm correspond with 
the results in determining the point where the intervals between the 
the sounds appear equal, and do not correspond with those where the 
intervals between the beginnings of the sounds appear equal, then 
the data concerning rhythm should be presented in a way which 
will make it clear what were the actual silent intervals between 
the sounds. Four subjects were tested in the above manner. It 
is unnecessary to insert at this point the detailed results of these 
tests, as the data are given in Chapter VI. In the case of three 
subjects, Dy, Ww, and Wr, it was found that the indifference point 
for rhythm corresponded with that for the comparison of the in- 
tervals from the end of one sound to the beginning of the following; 
while in the case of the fourth subject, Ws, the results on rhythm 
corresponded with the results on the comparison of intervals between 
the beginnings of the sounds. There is some ground, then, for the 
conclusion that usually the intervals which are most important 
in deciding what will be the nature of the rhythmical impression 
are the intervals between the end of one sound and the beginning 
of the following. But, in the case of Ws, there is no doubt that 
the rhythm judgment was more closely allied to the process of 
comparing the intervals between the beginnings of the sounds than 
to that of the comparison of the actual silent intervals. This 
conclusion is borne out by the following introspection: "I do not 
mean by intervals the time between the notes. The interval is 
the difference between the times when the two notes strike." Again, 
"Tones are more nearly of the same length as you approach the chang- 
ing point, the point where iambic changes to trochaic. The tones 
differ but the beats don't." Again, after an hour spent in attempting 
to compare the actual silent intervals between the sounds, "It 
is funny, because it is hard to catch the end of a sound and measure 
intervals in such a manner." The subject plays the piano and 
organ, and is the only piano player, in fact, the only practical mu- 
sician of any sort who took part in this investigation. In piano 
playing, of course, the beat producing any sound comes at the be- 
ginning of the sound, at least the greatest amount of tactual and 
kinaesthetic sensation would be received at the moment of pressing 
down the keys; moreover, in music, especially organ music, there 
are practically no silent intervals between the sounds. These 
considerations suggest that possibly the reason why the judgments on 
rhythm corresponded in this case with judgments on the intervals 



52 A QUANTITATIVE STUDY OF RHYTHM 

between the beginnings of the sounds was that there was in this 
case a motor performance coinciding with the beginning of the 
sounds, a performance which may have been absent, insignificant, 
or of a different character in the case of the other subjects. The 
importance of motor phenomena in the origin of the impression of 
rhythm has been sufficiently emphasized by previous authors. 

The results of this investigation show very plainly the significance 
of this question concerning the treatment of results. If we con- 
sider the two-membered group made up of two periods, each ex- 
tending from the beginning of one sound to the beginning of the 
next, it is true without exception that, when these periods are equal, 
the rhythm is heard as trochaic, and, providing these periods are 
kept equal, the rhythm becomes more and more trochaic as the 
ratio between the durations of the sounds increases. Thus, while 
in the case of the ratio of durations of 2 to 1, with a rather long 
absolute duration of the sounds, say a duration of the shorter sound 
equal to .40 second, we might expect to find the rhythm strongly 
iambic if the intervals between the sounds were equal, it would 
without question be heard as trochaic if the intervals between the 
beginnings of the sounds were equal. No subject was found, in this 
investigation of rhythms produced by variations in duration only, 
who in any case obtained iambic rhythm from any series of sounds, 
the intervals between the beginnings of which were equal. As long 
as the intervals between the beginnings of the sounds are equal, the 
rhythm becomes more and more trochaic with an increase in the 
ratio of the duration of the longer sound to that of the shorter. 
In some subjects, this increase in trochaism is very nearly 
proportional to the increase in the ratio of the durations, in 
others, it is less rapid. It seems, therefore, impossible to state 
the exact relation between the increase in the ratio of dura- 
tions and the increase in the temporal segregation of the group 
(in this case, the increase in trochaism) until the causes of individual 
differences can be quantitatively estimated. 



CHAPTER VI 
The Meaning of Rhythmical Grouping 

Every one is agreed today that the essential thing in the percep- 
tion of rhythm is the experiencing of groups. It is this experience 
of groups which distinguishes rhythm in the psychological sense of the 
word from rhythm in the sense of a regularly recurring event, such 
as the revolution of the earth about its axis. Thus, Meumann 
writes that the large number of well-trained observers, which he 
used in his investigations of rhythm, gave without exception the 
introspection that a subjective binding together of the impressions 
into a whole is inseparable from the simplest case of the perception 
of rhythm.^ Bolton found in the case of every one of thirty subjects 
that grouping was the irremissible sign of rhythm. And Miner 
states at the outset of his treatise on rhythm that he uses the word 
rhythm only in the sense of rhythmic grouping. 

But, while all recent writers on rhythm recognize that the ex- 
perience of groups is an essential factor in the experience known 
as the perception, or the feeling, of rhythm, there is wide diversity 
of opinion as to what we mean when we speak of the members of 
a series of impressions as being experienced in groups. I am un- 
able to judge just to what extent this diversity of opinion goes, 
because of the impossibility of ascertaining, in the case of many 
writers, whether they are referring to the experience of groups, 
or to the causes of this experience — whether they are describing 
a state of consciousness, or presenting a theory as to the origin 
of this state. To give an example, Bolton writes as follows: "The 
conscious state accompanying each wave of attention grasps to- 
gether or unifies all the impressions that fall within the temporal 
period of the wave."^ If he had written merely that the attention 
wave grasps the impressions together, one would believe that he 
was talking of the origin of the group experience; but when he says 
that it is the conscious state accompamjing the attention wave, it 
seems as though he was giving an introspective description of the 
group experience. Some of the interesting introspections reported 
by Bolton show the same ambiguity. He says of himself, "The 
puffs of a locomotive may now be grouped by two or three, but the 
association of the drive-wheel making one revolution to four sounds 
renders any other form of grouping than by four difficult."^ In 

^ Phil. Stud., 10, 271, 1894. 

^ Amer. Journ. of Psychol., 6, 220, 1894. 

^ Ibid., p. 205. 



54 A QUANTITATIVE STUDY OF RHYTHM 

this case, did the idea of the drive-wheel, Hke the conscious state 
accompanying an attention wave, grasp or tend to group four 
impressions together, and so produce a grouping or did the group 
experience consist merely in the simultaneous experience of the 
four impressions and the idea of the drive-wheel making one revo- 
lution? The introspections by his subjects are also often ambig- 
uous, for instance, the introspection of subject No. 12, of whom 
Bolton writes: "As to the nature of the group, the subject de- 
scribed his feelings as a tendency to go back when he had heard 
three or four clicks, as the case might be. He says he has a ' mouth- 
ful'— a unity — and when he has one, he seeks to get another."^ 
Another subject "noticed rhythms in the sound of mill wheels. 
When he gave his attention to these sounds he visualized a series 
of points on a line which he counted by four or two. When he 
was asked to count a series of dots he said they were divided off 
into twos by a bracket above them."^ In this case, the meaning 
of rhythmical grouping seems to be a grouping by brackets, the 
idea of bracketed points existing, apparently, more or less simul- 
taneously with the percepts of the mill sounds. Subject No. 7 
described some of his groupings as though grouping merely meant 
counting to eight. At times, Bolton seems to consider that the 
grouping is a temporal grouping, as when he writes as follows: 
"The weaker or less accented sounds seem to run together with the 
stronger, and to form organic groups which are separated from 
one another by intervals which are apparently longer than the 
interval which separates the individual clicks."^ But Bolton 
nowhere tries to show how "the conscious state accompanying 
each wave of attention" causes the impressions which it unifies 
to appear separated by shorter intervals than the intervals separa- 
ting two successive impressions which fall in different waves, i. e., 
in different groups. 

Meumann writes that for many observers the grouping was 
always temporal, a temporal holding together, in which the mem- 
bers of the groups appear to follow quicker upon each other, while 
between every two groups lies a pause.* He regards grouping as 
an intellectual act, which shows itself in the subordination of cer- 
tain impressions to others, for instance, the subordination of the 
more intense to the less intense. In verse, however, Meumann 

* Amer. Journ. of Psychol., 6, 196, 1894. 
^ Ibid., p. 198. 

^ Ibid., p. 204. 

* Phil. Stud., 10, 283 and 304, 1894. 



THE MEANING OF RHYTHMICAL GROUPING 55 

says that grouping may be conditioned entirely by meaning.^ In 
this case, then, rhythmical grouping means logical unity. 

The position of McDougall seems to be that a rhythmical group 
is an awareness of a temporal segregation of the impressions. "The 
whole group of elements constituting the rhythmic unit is present 
to consciousness as a single experience; the first of its elements 
has never fallen out of consciousness before the final member ap- 
pears, and the awareness of intensive differences and temporal 
segregation is as immediate a fact of sensory apprehension as is 
the perception of the musical qualities of the sounds themselves. "^ 
At other times, he seems to regard the grouping as the experiencing 
of impressions along with the experience of an ideal form, or 
''Gestaltsqualitat," as when he speaks of the experience of rhythm 
as being supported by the conception of an ideal form which the 
series of stimuli fulfils.^ He says further that the synthesis of 
elements may be mediated by changes in the ideal significance 
and relation of the various members; also by movements of the 
head, jaw, throat, eyes, or by muscular strain;^ but he does not 
show how these changes in ideal significance or these movements 
of various parts of the body produce temporal segregation. 

The view has been put forth by several recent authors that the 
rhythmical group consists in the experience of muscular strains 
along with the experience of the other impressions, such as 
a series of sounds. According to this view, a rhythmical group 
may be thought of as a series of impressions strung together by 
one longer state of strain, much like fish strung together on a string; 
as when Miner speaks of the separate sensations from the external 
world as being strung in groups. Stetson represents this view 
when he says that "the continuity of the rhythmic series, whereby 
all the beats of a period seem to belong to a single whole, is due to 
the continuity of the muscle sensations involved and the continuous 
feeling of slight tension between the positive and negative muscle 
sets;^ nowhere within the period does the feeling of strain die 
out." Similarly, Miner says that "feeling the groups to be units 
is an illusion due to the presence of movement or strain sensations 
along with the sensations that are grouped." "These kinaes- 
thetic sensations provide the factor by which unit sensations ap- 

^ Phil. Stud., 10, 396, 1894. 

^ Harvard Psychol. Stud., i, 1903 and Monog. Sup. Psychol. Rev., 4, 322, 
1903. 

3 Ibid., p. 468. 

* Ibid., p. 343. 

^ Harvard Psychol. Stud., i, 1903 and Monog. Sup. Psychol. Rev., 4, 455, 
1903. 



56 A QUANTITATIVE STUDY OF RHYTHM 

pear bound into groups."* In speaking of visual rhythms, how- 
ever, he says: "The units in the group seem crowded closer to- 
gether and a longer interval appears before the next group starts."^ 
This indicates that the grouping is temporal. We might expect, 
therefore, some attempt on the part of the author to show that 
strain sensations, which are said to hold the impressions together, 
cause an underestimation in time; he makes no such attempt. And 
while it might possibly be held that strain sensations and the per- 
ception of time are identical, such a theory would rather lead us to 
expect the strain sensations to force the separate impressions apart 
from each other rather than hold them together, because accord- 
ing to any theory which makes strain sensations the basis of the 
perception of time, the greater the strain sensations during any 
interval, the greater the apparent duration of that interval. 

As I have already stated, in the present research in all the work 
done on the effect of duration and intensity on rhythm, the sub- 
jects were instructed to judge concerning the rhythm, and noth- 
ing was said to them concerning intervals. In order to obtain 
some idea of the nature of the rhythmical group, however, after 
the work on rhythm was all over, I carried on some investigations 
on the effect of the same variations in intensity and duration that 
I had used on rhythm on the apparent length of the intervals. 
The indifference points of which I have spoken in the chapters 
on intensity and duration were indifference points for rhythm. 
This rhythm indifference point should not be confused with what 
is spoken of as the indifference point in the comparison of inter- 
vals. The instructions given the subject and the experimental 
procedure used in obtaining the various rhythm indifference points 
have already been described. The procedure for obtaining the 
point at which all the intervals appear equal, that is, for obtaining 
the indifference points in the comparison of intervals, was in every 
respect the same as that used in the study of the indifference point 
for rhythm; but the instructions given the subject were different. 
In obtaining the indifference point for intervals, the subject was 
asked to ignore the rhytiim, and (every second sound being more 
intense or longer) to judge merely whether the interval preceding 
the louder or the longer sound was greater, equal, or less than that 
following it. 

When there is much of a difference in the duration of the sounds, 
the subject's idea of what is meant by intervals will evidently have 
a great influence on the results obtained. If the word interval 

^ Monog. Sup. Psychol. Rev., 5, No. 4, pp. 2, 20, 1903. 
2 Ibid., p. 5.5. 



THE MEANING OF RHYTHMICAL GROUPING 57 

means to the subject the time between the beginnings of the sounds, 
the indifference point for the comparison of intervals will be quite 
different from what is obtained when the subject is asked to con- 
sider the intervals which exist between the sounds. No instruc- 
tions were at first given to any of the subjects concerning what 
they should consider to be the interval, as it was desired to see 
what they did naturally. Rather curiously none of the subjects 
seemed in any doubt as to what intervals to compare. The possi- 
bility that interval might mean anything but one thing never oc- 
curred to them, unless it was pointed out to them; and yet one of 
the subjects, Ws, started in to compare the intervals between the 
beginnings of the sounds, while three others compared the inter- 
vals of silence between the sounds. In all four subjects the results 
obtained by the comparison of intervals, leaving the subject to 
follow his own ideas concerning the meaning of intervals, gave as 
the indifference point for the comparison of intervals in any given 
series of sounds practically the same point which had been pre- 
viously obtained as the rhythm indifference point for the same 
series. As long as the subjects compared intervals in the way 
which seemed most natural to them, their results for the interval 
indifference point and the rhythm indifference point corresponded 
very closely; but when subject Ws was asked to compare actual 
intervals, results were obtained very different from the results ob- 
tained on the rhythm indifference point; and when subjects Dy, 
Ww, and Wr compared the intervals between the beginnings of 
the sounds, the interval indifference points thus obtained diverged 
widely from the corresponding rhythm indifference points. By 
"before" and "after" in the following table is meant, as usual, 
the actual intervals of silence before and after the longer sound. 







TABLE XII 








Duration of shorter sound (constant) = o 


13 second. 


Total duration of 1 


= 1.5 seconds. 
















r sounc 


Intervals at indifference p 


oint 


N'&turc of iii~ 


Subject I,onge 


I Before 


After 


difference point 


Wr 


60 


•44 




33 




rhythmic 


Wr 


60 




46 




31 




silence 


Wr 


60 




55 




22 




beginnings 


Ws 


60 




54 




23 




rhythmic 


Ws 


28 




55 




54 




rhythmic 


Ws 


28 




54 




55 




beginnings 


Ws 


28 




45 




64 




silence 


Ww 


37 




51 




49 




rhythmic 


Ww 


37 




51 




49 




silence 


Ww 


37 




5§ 




.42 




beginnings 



58 



A QUANTITATIVE STUDY OF RHYTHM 



Results are presented for three different sorts of indifference point, 
the iambic-trochaic indifference point, the indifference point for 
the comparison of the actual intervals of silence between the end 
of one sound and the beginning of the following sound, and the 
indifference point for the comparison of the intervals between the 
beginnings of the sounds. These three sorts of indifference point 
are referred to in the column headed ''nature of indifference point," 
as "rhythmic," "silence," and "beginnings." 

The comparison of intervals thus affords a method of determin- 
ing what intervals are the most significant in rhythm, for, evidently, 
if comparing intervals in one way gives results the same as obtained 
in judging rhythm, while comparison in a second way gives widely 
different results, the process of judging of the rhythm must be more 
closely allied to the first way of comparing intervals than to the 
second. 

TABLE XIII 





] 


ntensity of 


weaker sc 


und (cc 


)nstant) = 


= 24 feet. 










o-a 


v> 

3 V 

















15' 


«J3 
M 




Indifference point 
for interval 
comparison 




Indifference point 

for rhythm 
(iambic-trochaic) 




3 

CO 


N 


M V Before After Before After 


N 


MV 


Ww 


136 


■13 


19 


■037 


•57 


.66 


.55 .68 


21 


.015 


Ww 


136 


.22 


15 


.026 


•94 I 


. II 


90 I 


16 


10 


.022 


Ww 


136 


■31 


10 


■033 I 


31 I 


.56 I 


29 I 


58 


10 


.021 


Ww 


136 


•45 


16 


.028 I 


92 2 


17 I 


91 2 


19 


14 


.041 


Ww 


136 


68 


16 


■055 2 


93 3 


22 2 


97 3 


18 


6 


.104 


Ww 


136 


07 


14 


.008 


30 


32 


26 


36 


10 


.010 


Ww 


136 


09 


H 


.Oil 


36 


46 


37 


46 


12 


.006 


Ws 


28 


13 


II 


.018 


62 


62 


61 


62 


20 


.022 


Ws 


32 


13 


10 


.018 


60 


63 


59 


64 


20 


.012 


Ws 


70 


13 


10 


.018 


60 


63 


58 


65 


20 


.021 


Ws 


196 


13 


15 


.010 


60 


63 


57 


66 


56 


■02 5 


Ws 


300 


13 


10 


.012 


58 


65 


57 


66 


41 


.021 


Ws 


1 100 


13 


10 


.015 


56 


67 


55 


68 


38 


.018 


Wr 


136 


13 


18 


.010 


58 


65 


56 


68 


10 


.010 


Wr 


1 100 


13 


II 


.016 


54 


69 


51 


72 


17 


.016 


Dy 


136 


13 


19 


.019 


58 


65 


51 


72 


15 


.016 


Dy 


420 


13 


14 


.024. 


48 


75 


50 


73 


10 


.014 



The results obtained on the indifference point for the comparison 
of intervals, are shown in Tables XIII and XIV, in which are also 
given the results on the rhythm indifference point for the same 
ratios of intensities and durations of the sounds. In Table XIII, 
results are presented for sound series in which all the sounds of a 
series were of the same length but every other sound the louder. 



THE MEANING OF RHYTHMICAL GROUPING 



59 



In Table XIV results are presented for series in which the sounds 
were of equal loudness but every other sound the longer. In the 
columns headed "before" and "after" are given the duration, 
at the indifference point, of the intervals before and after the longer 
or the more intense sound. 

TABLE XIV 
Intensity of all sounds = 24 feet. 





15' 


<n 




01 

u 


Indifference point 
for interval 
comparison 








[ndifference point 

for rhythm 
iambic-trochaic) 






N MV Before After 


Before 


After 


N 


m'v 


Ws 


28 


13 


10 .oop 


53 


56 


-55 


■54 


24 


.020 


Ww 


37 


13 


15 -015 


51 


49 




51 




49 


15 


.016 


Ww 


61 


22 


12 .022 


76 


90 




75 




91 


12 


.018 


Ww 


86 


30 


8 .028 


98 I 


34 


I 


GO 


I 


32 


14 


.028 


Ww I 


23 


43 


6 .033 I 


53 I 


80 


I 


47 


I 


86 


6 


.028 


Ww I 


65 


65 


6 . 030 2 


35 2 


65 


2 


21 


2 


79 


6 


■037 


*Dy 


37 


13 


12 .019 


43 


57 




40 




60 


10 


■045 


Wr 


19 


13 


10 .010 


58 


60 




58 




60 


II 


.Oil 


Wr 


25 


13 


9 .010 


54 


58 




54 




58 


16 


.018 


Wr 


31 


13 


15 .077 


52 


54 




53 




53 


17 


.oil 


Wr 


43 


13 


15 .018 


47 


45 




51 




43 


17 


.023 


Wr 


60 


13 


13 .027 


46 


31 




44 




33 


19 


.024 


*In 


this ca 


36 the 


longer sound ha 


d an int 


ensity 


of 


70 


feet. 









The results show that the absolute and relative durations of the 
intervals at the iambic-trochaic indifference point are almost the 
same as when the two intervals (the intervals before and after the 
louder or the longer sound) appear to be equal in length. The 
rhythm indifference points and the interval indifference points are 
almost, but not entirely, identical. If we term the differences 
in the durations of the intervals before and after the louder or the 
more intense sound constant errors, then it may be said that con- 
sidering both Tables XIII and XIV together, most frequently the 
constant errors are less when the subject judges intervals than 
when he judges rhythm. In the case of series in which the sounds 
present only duration differences, however, there seems to be no 
reliable difference, as in Table XIV the constant errors for rhythm 
are in five cases greater, in three cases the same, and in four cases 
less than the constant errors for comparison of intervals. In Table 
XIII, however, it is fairly evident, perhaps, that the constant er- 
rors due to differences in the intensity of the sounds are usually 
slightly greater for the rhythm indifference points than for the 
interval indifference points, as the constant errors for the rhythm 



60 A QUANTITATIVE STUDY OF RHYTHM 

judgments are greater than those for the interval judgments in 
fourteen cases and less in three cases. The two sets of results 
are not exactly identical, but they are so close to it that the con- 
clusion seems fairly safe that they are the result of mental opera- 
tions M^hich have about the same basis. In other words, the state- 
ment by the subject, that certain sounds form an iambic group, 
is equivalent to the statement that he has perceived a shorter in- 
terval before the louder or the longer sound than after it, and the 
statement that the sounds form a trochaic group means that the 
subject has perceived a shorter interval after the louder or longer 
sound than before it. This is the equivalent of saying that the 
experiencing of a rhythmical group is an experiencing of temporal 
relations, that the meaning of rhythmical grouping is temporal 
segregation. This conclusion in no way implies that the subject 
makes judgments concerning intervals whenever he hears rhythm, 
but merely that he perceives these intervals. Nor does it imply 
that in comparing intervals one necessarily judges of rhythm; 
nor possibly, even, that he has experienced rhythm, since it has 
never been proved that the immediate experience of a temporal 
grouping is of itself all that is necessary for the experience of rhythm. 
In the hope of making clearer the relation between the percep- 
tion of a temporal grouping and the perception of rhythm, I cite 
a few introspections. They are quoted literally, and are to be 
treated as data requiring interpretation, and not accepted at their 
face value. Subject Br: "The rhythmical grouping seems a tem- 
poral grouping without attention being paid to the temporal rela- 
tions at the time the rhythm is heard." Subject Wr was asked, 
after his first hour of work in comparing intervals, which inter- 
vals he had been comparing, the intervals between the beginnings 
of the sounds or the intervals of silence between the sounds. He 
replied that he did not know which he had been doing. Subject 
Dy: "Rhythm is not always present in comparing intervals, but 
it sometimes comes. At such times it seems difficult to judge in- 
tervals without the rhythm being the predominating factor. The 
rhythm I notice in comparing intervals is not always the same as 
when I put my attention on the rhythm itself, e. g., ii I get a tro- 
chaic rh5^thm while judging intervals, and then turn my attention 
to the rhythm I sometimes get iambic instead. An iambic never 
changed to a trochaic in this way." Ws: "Rhythm becomes ap- 
parent to the ear by unequal lengths of sound. The short inter- 
vals come between the sounds which make the rhythm and the 
long intervals between the groups of sounds." At another time, 
"When the loudest and longest note comes first, I put down tro- 



THE MEANING OF RHYTHMICAL GROUPING 61 

chaic; but when it comes at the end of each rhythm, I put down 
iambic. I do not notice the length of intervals especially." Same 
subject, summer, '07: "When there is absolutely no difference in 
the length of intervals, the rhythm is trochaic." Summer, '08: "In 
comparing the intervals, the intervals are equal in iambic rhythm." 
After an experiment in which this same subject was asked to judge 
in the case of each sound series, first, whether the rhythm was 
iambic, doubtful, or trochaic, and second, whether the intervals 
were equal or unequal, and if unequal whether the interval before 
or after the louder sound was the longer. "In keeping a record of 
both rhythms and intervals I went through two different processes 
of judgment; the two judgments did not seem alike. I had never 
before realized that the doubtful rhythms were those where the 
intervals are equal." 

The very slight difference already noted as existing between the 
interval indifference point and the indifference point for rhythm 
does not seem large enough nor reliable enough to invalidate the 
conclusion that the grouping referred to when we speak of rhythmical 
grouping is a temporal grouping. What slight tendency there is 
for constant errors (in the sense used above) to be smaller when the 
subject judges of intervals than when he judges of rhythm seems 
to me to indicate merely the effect of the different attitude on the 
part of the subject in the two cases, the difference being, to judge 
from my own introspection, that, in comparing intervals, both 
sounds are about evenly attended to, there is a comparatively 
continuous strain, while in judging rhythm, some of the sounds, 
e. g., the more intense, are attended to more than others. Such a 
view enables us to understand the statement of Meumann that his 
subjects only judged the interval between the groups as longer 
than the intervals within the group, when they did not set out espe- 
cially to judge intervals ("Wenn sie nicht darauf ausgehen, die 
Intervalle bewusst zu vergleichen."^) 

We are now in a position to understand in what consists the 
grouping effect of duration and intensity, and consequently their 
rhythmical effect in so far as rhythm consists in grouping. A regu- 
larly recurring more intense sound has the effect of increasing the 
temporal value of the interval preceding it as compared with that 
following it. This means that if the intervals are equal the sounds 
appear temporally grouped in such a way that the interval pre- 
ceding the more intense sound separates the group, that is, that 
the sounds, if heard as rhythm, produce the impression of trochaic 
rhythm. This effect of the more intense sound may therefore be 

^ Phil. Stud., 10, 304, 1894. 



62 A QUANTITATIVE STUDY OF RHYTHM 

spoken of as trochaic. The regularly recurring more intense sound 
continues to exert this effect even though, because of an objective 
shortening of the interval preceding the more intense sound, the 
rhythm is heard as iambic. The more intense sound is exerting 
an influence towards trochaism even when it occurs in an iambic 
rhythm. Similarly, if the distinction between the duration effect 
per se and the intensity effect of an increase in duration be admitted, 
the results obtained on duration show that a regularly recurring 
shorter sound also exerts a trochaic influence, and this, too, whether 
the rhythm is heard as iambic or trochaic. In referring to the 
trochaic effects of the shorter and the londer sounds, I am speak- 
ing relatively. I mean that the effect of the regularly recurring 
shorter or louder sound is trochaic compared to the effect of the 
weaker or the longer sound. It is equally true, of course, if we are 
speaking relatively, to say that an increase in duration or a decrease 
in intensity exerts an iambic effect, or causes an underestimation, 
of the interval preceding the longer or the weaker sound. 



SUMMARY 

It is possible to pass from one rhythmical grouping to another 
by changing the relative duration of the intervals between the 
sounds. Thus, a trochaic rhythm, that is, one that is composed 
of groups of two sounds each, the louder sound beginning the group, 
may be changed to an iambic rhythm, one in which the louder 
sound ends the group, by increasing the interval immediately 
following the louder sound or by decreasing the interval immediately 
preceding it. Similarly, a rhythmical group which begins with a 
sound longer than the other sounds of the group may be changed 
to one in which the longer sound ends the group by increasing the 
interval immediately following the longer sound, or by decreasing 
all the other intervals. As we pass in this way from rhythmical 
groups beginning with the louder or longer sound to rhythmical 
groups ending with that sound, we pass through a zone where the 
tendency towards the two forms of grouping is equally strong. 
The middle point of this zone may be termed the rhythm indifference 
point. If the rhythm indifference point occurs where all the inter- 
vals are objectively equal, then any differences which may exist 
in the objective duration or loudness of the sounds are obviously 
not exerting any effect towards grouping. But if, when the intervals 
between all the sounds are objectively equal, grouping is still per- 
ceived, the grouping must be regarded as brought about by other 
factors than objective differences in the intervals. The amount 
of this grouping effect can be determined by finding out what change 
from objective equality of intervals is necessary in order to cause 
the grouping to disappear, that is, by ascertaining the amount by 
which one of the intervals has to be lengthened or shortened with 
respect to the others in order to arrive at the rhythm indifference 
point. We may say, therefore, that the influence towards rhyth- 
mical grouping exerted by factors other than objective differences 
in the intervals, that is, by such factors as recurrent differences 
in accent, duration, pitch, etc., is measured by the difference, at the 
rhythm indifference point, between the external and internal in- 
tervals of the group. 

When the intervals are equal, and every second stimulus the 
stronger, the rhythm is trochaic, and when every third is the stronger, 
dactylic. That is, a regularly recurring difference in intensity 
exerts a tendency towards rhythmical groups with the more intense 
sound at the beginning. In other words, accenting certain sounds 
of a series has the same effect on the position of those sounds within 



64 A QUANTITATIVE STUDY OF RHYTHM 

the rhythmical group as objectively increasing the interval preceding 
them. For instance, if the sounds corresponding to the odd numbers 
of an equally spaced series be accented, those sounds appear to 
begin the groups; and similarly, without accenting them, but by 
sufficiently increasing the interval immediately preceding them, 
they may be made to begin the groups. 

This trochaic or dactylic effect caused by an increase in the rela- 
tive intensity of every second or every third sound, that is, the 
tendency of the accented sound to begin the rhythmical group, 
may be measured by the amount by which the interval immediately 
following the stronger stimulus has to be increased in order to arrive 
at the rhythm indifference point: and in this way the effect on 
rhythm of variations in the relative intensity of the sounds may be 
studied. With an increase in the ratio of the intensity of the louder 
sound to that of the weaker, there is an increase, first rapid and then 
slow, in the tendency of the more intense sound to begin the group. In 
other words, with equal intervals, as the difference in intensity 
between the louder and weaker sounds increases, the intensity of 
the weaker sound remaining constant, the rhythm becomes more 
and more trochaic, if composed of two-membered groups, or more 
and more dactylic, if composed of three-membered groups. 

I have stated that the rhythmical effect exerted by regularly 
recurrent accents is measured by the difference between the in- 
ternal and external intervals of the group at the indifference point. 
This is an absolute measure. By dividing this by the total duration 
of one measure we get a relative measure, that is, a measure of the 
rhythmical effect relative to the total duration of one measure. 
In comparing the rhythmical effect of intensity, duration, etc., in 
sound series which are run off at different rates, the relative measure 
is what should be taken into consideration. The rhythmical effect 
of any given ratio of intensities between the louder and weaker 
sounds, relative to the rate at which the series is run off, remains 
constant, in two-group rhythms, for rates varying approximately 
from one to four seconds for one measure, usually shows a marked 
decrease by the time a rate of seven seconds for one measure is 
reached, but in some cases does not entirely disappear at a rate of 
ten seconds for one measure. 

Measurements of the rhythmical effect of changes in the relative 
and absolute duration of sounds, made by the same method as that 
used in the case of intensive differences, lead to the following gen- 
eralizations. With an increase in the ratio of the duration of the 
longer sound to that of the shorter, there is an increase in the tendency 
of the longer sound to end the group or a decrease in its tendency to 



SUMMARY 65 

hegin the group. When the ratio of the duration of the longer sound 
to the duration of the shorter is small, that is, when there is not 
much difference in the duration of the sounds, and when, further, 
the absolute duration of the sounds is also small, the longer sound 
tends to begin the group. A small regularly recurrent increase in 
duration, then, may have the same effect, providing the absolute 
duration of the sounds is small, as a regularly recurrent difference 
of accent. But whereas an increase in relative intensity has the 
effect of increasing the tendency of the accented sound to begin 
the group, an increase in relative duration has the effect of decreasing 
the tendency of the longer sound to begin the group, and often 
results in an exceedingly strong tendency on the part of the longer 
sound to end the group. 

If all intervals are kept equal, and every second sound is some- 
what longer, we may have a trochaic rhythm. This seems at first 
sight to indicate that the effect of a slight increase in duration is 
a trochaic tendency whereas the effect of a considerable increase 
is an iambic tendency. It is necessary, however, to make a dis- 
tinction between duration per se and the increase in apparent in- 
tensity of a stimulus due to an increase in duration. The apparent 
intensity increases as the duration, at first fast but later very slowly. 
The difference in the apparent intensity of a sound one second in 
duration and of a sound two seconds in duration is small, very 
small compared to the difference in duration. The difference in 
apparent intensity of a sound one-fiftieth of a second in duration 
and a sound two-fiftieths of a second in duration is very great, 
even in comparison with the increase in apparent duration. As 
we increase the duration of every second sound, therefore, we have 
two separate and antagonistic effects to keep in mind. First, 
there is an increase in the tendency of the longer sound to begin 
the group, due to the effect of duration on apparent intensity: 
second, there is a much more rapid increase in the tendency of the 
longer sound to end the group, due to the effect of an increase in 
duration per se. The increase in the second tendency is so much 
faster than that in the first, that the second may overcome the first, 
when the increase in duration is great, even though the first ten- 
dency may have been the stronger when the increase in duration 
was small. 

The difference in the apparent intensity between two sounds 
due to a difference in their duration decreases, compared to the 
difference in apparent duration, as the absolute duration of the 
sounds increases. As the absolute duration of the sounds is in- 
creased, then, the trochaic effQct of duration due to the effect of 



66 A QUANTITATIVE STUDY OF RHYTHM 

duration on intensity should decrease relatively to the iambic effect 
of duration as such. And measurements show that, in fact, with a 
constant ratio between the durations of the sounds, as their absolute 
duration increases, there is a decrease in the tendency of the longer 
sound to begin the group or an increase in its tendency to end the group. 
The effect of both intensity and duration in rhythm may be 
generalized as follows. If every second or third sound is made 
more intense or is made shorter, the effect on grouping is the same 
as if the interval immediately preceding that sound were increased 
relative to the other intervals. The effect of the more intense 
sound, when all the sounds are of equal duration, or of the shorter 
sound, when all the sounds are of equal intensity, is a relative over- 
estimation of the interval preceding the more intense or the shorter 
sound. There is an objection to speaking of an overestimation 
of any interval, however, in that the subjects in the experiments 
so far considered were not estimating intervals, but were judging 
rhythm. But when the subjects were instructed to estimate in- 
tervals, it was found that, in fact, the interval preceding the regularly 
recurrent more intense sound or the regularly recurrent shorter sound 
is relatively overestimated. The rhythm indifference point and the 
indifference point for the estimation of intervals are almost, though 
not exactly, identical. This close correspondence between the 
rhythmical grouping and the temporal grouping, or rather this 
correspondence in the points where both disappear, indicates that 
rhythmical grouping is a temporal grouping; that is, that rhythmical 
grouping is determined by the duration of the subjective intervals, 
not by the objectively measurable intervals, but by the subject's 
consciousness of these intervals, that is, by the intervals considered 
as mental magnitudes. 



VITA 

The author was born in ChilHcothe, Ohio, February 25, 1883. 
Matriculated in the University of Michigan 1900; A. B., 1904. 
Matriculated in the University of Paris, Faculte des Lettres, 1904. 
Demonstrator in Experimental Psychology, Princeton University, 
1907. Lecturer in Psychology in Columbia University, 1907-1909; 
appointed Tutor, 1909. For inspiration and encouragement the 
author is indebted to Professors Pillsbury and liombard and Dr. 
Shepard of the University of Michigan, to Professor G. Dumas and 
■the late Professor Egger of the Sorbonne, Paris, and to Professors 
Cattell, Lee, Woodworth, and Thorndike of Columbia University. 



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